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**Simulated Annealing for Traveling Salesman Problem**
Simulated annealing is a local search algorithm used to find satisfactory solutions to complex problems with numerous permutations or combinations. It operates by decreasing temperature according to a schedule, allowing it to transition from random to improved solutions.
**Key Characteristics:**
* Uses decreasing temperature to control the probability of accepting inferior solutions
* Can find satisfactory solutions quickly without requiring excessive memory
* Not guaranteed to find optimal solutions, but can produce good results
**Traveling Salesman Problem (TSP):**
The TSP involves finding the shortest route that visits a set of cities and returns to the starting city. This problem has a large number of permutations, making it challenging to solve optimally.
**Example Problem:**
A TSP with 13 cities has 479,001,600 possible permutations, making it impractical to test every permutation. The goal is to find the route with the shortest total distance that includes all cities and starts and ends in the same city.
**Solution Approach:**
1. **Initial Solution:** The initial solution can be selected at random, using distance weights, or by choosing the shortest distance in each step.
2. **Simulated Annealing:** The simulated annealing algorithm is applied to the initial solution to find improved solutions. It uses a schedule to control the temperature and the probability of accepting inferior solutions.
**Code Implementation:**
The solution is implemented in Python, using a simulated annealing algorithm to find the shortest route. The code includes functions for:
* Generating initial solutions
* Mutating solutions
* Calculating distances
* Applying the simulated annealing algorithm
**Output:**
The output includes the best solutions found using different approaches:
* Best solution by distance: 8,131 miles
* Best random solution: 11,324 miles
* Best random solution with weights: 8,540 miles
* Simulated annealing solution: 7,534 miles
The simulated annealing algorithm produces a solution that is better than the initial solution, demonstrating its effectiveness in finding satisfactory solutions to complex problems.
**Simulated Annealing Algorithm:**
The simulated annealing algorithm works as follows:
1. Initialize the temperature and the current solution
2. Generate a new solution by mutating the current solution
3. Calculate the change in distance between the new and current solutions
4. If the new solution is better, accept it as the current solution
5. If the new solution is not better, accept it with a probability based on the temperature and the change in distance
6. Decrease the temperature according to the schedule
7. Repeat steps 2-6 until the temperature reaches zero or a stopping criterion is met
The algorithm uses a schedule to control the temperature, allowing it to transition from exploring the solution space to exploiting the best solutions found. The probability of accepting inferior solutions decreases as the temperature decreases, ensuring that the algorithm converges to a good solution.
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CC-MAIN-2024-42/segments/1727944253762.73/warc/CC-MAIN-20241011103532-20241011133532-00145.warc.gz
|
annytab.com
|
en
| 0.735634
| 2024-10-11T12:21:38
|
https://www.annytab.com/simulated-annealing-search-algorithm-in-python/
| 0.724939
|
1 Groups and Homomorphisms
1.3 Cyclic Groups
A cyclic group is a group of the form {e, a, a^2, ..., a^(n-1)}, where a^n = e. For example, the group of all rotations of a triangle can be represented as {e, r, r^2} with r^3 = e, where r = rotation by 120°.
Definition (Cyclic Group C_n): A group G is cyclic if every element is some power of a, denoted as (∃a)(∀b)(∃n ∈ Z) b = a^n. Such an element a is called a generator of G. The cyclic group of order n is denoted as C_n.
Examples:
(i) The set of integers Z is a cyclic group with generator 1 or -1, known as the infinite cyclic group.
(ii) The set ({+1, -1}, ×) is a cyclic group with generator -1.
(iii) The set (Z_n, +) is a cyclic group with all numbers coprime with n as generators.
Notation: Given a group G and an element a ∈ G, the cyclic group generated by a is denoted as <a>, representing the subgroup of all powers of a. It is the smallest subgroup containing a.
Definition (Order of Element): The order of an element a is the smallest integer n such that a^n = e. If no such n exists, the element a has infinite order, denoted as ord(a).
Note that the term "order" has two different meanings: the order of a group and the order of an element. The relationship between these two concepts is established by the following lemma:
Lemma: For an element a in a group, ord(a) = |<a>|.
Proof: If ord(a) = ∞, then a^n ≠ a^m for all n ≠ m, implying that |<a>| = ∞ = ord(a). Otherwise, suppose ord(a) = k, then a^k = e. The cyclic group <a> can be expressed as {e, a^1, a^2, ..., a^(k-1)}, as higher powers of a will loop back to existing elements, and there are no repeating elements in the list.
Proposition: Cyclic groups are abelian.
Definition (Exponent of Group): The exponent of a group G is the smallest integer n such that a^n = e for all a ∈ G.
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CC-MAIN-2023-06/segments/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00498.warc.gz
|
srcf.net
|
en
| 0.87993
| 2023-02-09T09:51:20
|
https://dec41.user.srcf.net/h/IA_M/groups/1_3
| 0.997593
|
In the context of Lie groups, a left-invariant metric can be defined using a scalar product on the Lie algebra. Given a Lie group G and a scalar product on its Lie algebra , a left-invariant metric on G can be defined using left translations. For a path in G, the tangent vector can be mapped to using left translations.
If is a geodesic for the metric , it satisfies Arnold's equation:
where is defined as , and are the structure constants of . The structure constants are defined as , where form a basis for .
Arnold's equation is a system of nonlinear ordinary differential equations with constant coefficients. Solving this equation is only half of the problem of finding a geodesic. Once is known, a linear differential equation with variable coefficients must be solved to find .
For the rotation group O(n), the equations of motion for a free rigid body are completely integrable. In general, there are two quadratic constants of motion: "kinetic energy" and "square of the angular momentum". The kinetic energy is a constant, independent of . This can be shown by differentiating the kinetic energy with respect to :
Substituting Arnold's equation into this expression yields:
The result is zero because is antisymmetric in , while is symmetric.
The second quadratic invariant can be derived using the Ad-invariant scalar product . The Ad-invariance of implies a relation between the matrix and the structure constants:
Differentiating this expression at yields:
Setting and multiplying both sides by yields:
The angular momentum is defined as , a covector in the dual of . The second conservation law states that the square of the angular momentum evaluated with the Ad-invariant metric is constant:
To verify this, we differentiate and use Arnold's equation:
According to , is antisymmetric in , while the product is symmetric, resulting in zero.
In future posts, we will apply similar reasoning to the case of the Lorentz group O(2,1) in 2+1 dimensions, building on the results for the rotation group in three dimensions and the rigid body.
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CC-MAIN-2024-38/segments/1725700651318.34/warc/CC-MAIN-20240910192923-20240910222923-00313.warc.gz
|
arkadiusz-jadczyk.eu
|
en
| 0.897001
| 2024-09-10T19:44:14
|
https://arkadiusz-jadczyk.eu/blog/category/eulers-equations/
| 0.998226
|
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. The momentum of an object is defined as the product of its mass (m) and velocity (v): p = mv.
The unit of momentum is the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second, then the momentum is in kilogram meters per second (kg⋅m/s). Momentum is conserved in a closed system, meaning that if a closed system is not affected by external forces, its total linear momentum does not change.
Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame, it is a conserved quantity. The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects, such as a rocket ejecting fuel or a star accreting gas.
In relativistic mechanics, momentum is defined as the product of the rest mass (m0) and the velocity (v) of an object: p = γm0v, where γ is the Lorentz factor. The relativistic energy-momentum equation is given by E^2 = (pc)^2 + (m0c^2)^2, where E is the total energy of the object, p is its momentum, and c is the speed of light.
In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. The momentum operator is given by p = -iℏ∇, where ℏ is the reduced Planck constant and ∇ is the gradient operator.
The concept of momentum has a long history, dating back to the ancient Greeks. The Greek philosopher Aristotle believed that everything that is moving must be kept moving by something. The concept of momentum was later developed by medieval philosophers, such as John Philoponus and Ibn Sīnā, who proposed that an impetus is imparted to an object in the act of throwing it.
In the 17th century, the concept of momentum was further developed by scientists such as René Descartes, Christiaan Huygens, and John Wallis. Descartes believed that the total "quantity of motion" in the universe is conserved, while Huygens formulated the correct laws for the elastic collision of two bodies. Wallis published the first correct statement of the law of conservation of momentum in his 1670 work, Mechanica sive De Motu, Tractatus Geometricus.
Today, the concept of momentum is a fundamental principle in physics and engineering, and is used to describe a wide range of phenomena, from the motion of objects on Earth to the behavior of particles in high-energy collisions. The law of conservation of momentum is a fundamental principle in physics, and is used to describe the behavior of closed systems, where the total momentum remains constant over time.
Momentum is a measurable quantity, and the measurement depends on the frame of reference. For example, if an aircraft of mass m kg is flying through the air at a speed of 50 m/s, its momentum can be calculated to be 50m kg.m/s. If the aircraft is flying into a headwind of 5 m/s, its speed relative to the surface of the Earth is only 45 m/s, and its momentum can be calculated to be 45m kg.m/s.
In a closed system, the total momentum remains constant over time. This is known as the law of conservation of momentum. The law of conservation of momentum can be used to determine the momentum of an object after a collision, and is a fundamental principle in the study of collisions and projectile motion.
The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, and velocities v1 and v2, the total momentum is given by the equation: p = m1v1 + m2v2. The momenta of more than two particles can be added more generally with the following equation: p = ∑mi vi.
A system of particles has a center of mass, a point determined by the weighted sum of their positions. If one or more of the particles is moving, the center of mass of the system will generally be moving as well. The momentum of the system is given by the equation: p = mv, where m is the total mass of the system and v is the velocity of the center of mass.
The concept of momentum is also used in the study of continuous systems, such as electromagnetic fields, fluid dynamics, and deformable bodies. In these systems, a momentum density can be defined, and a continuum version of the conservation of momentum leads to equations such as the Navier-Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.
In conclusion, momentum is a fundamental concept in physics, and is used to describe the motion of objects and the behavior of closed systems. The law of conservation of momentum is a fundamental principle in physics, and is used to determine the momentum of an object after a collision. The concept of momentum has a long history, dating back to the ancient Greeks, and has been developed over time by scientists such as Descartes, Huygens, and Wallis. Today, the concept of momentum is a fundamental principle in physics and engineering, and is used to describe a wide range of phenomena, from the motion of objects on Earth to the behavior of particles in high-energy collisions.
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CC-MAIN-2022-21/segments/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00145.warc.gz
|
knowpia.com
|
en
| 0.828477
| 2022-05-20T04:09:44
|
https://www.knowpia.com/knowpedia/Momentum
| 0.922823
|
Pierre de Fermat's method of infinite descent is a key concept in Number Theory, beautifully illustrated by the proofs of two propositions. The first proposition states that there are no integer solutions of x^4 + y^4 = z^2. To prove this, suppose there are integers x, y, z such that x^4 + y^4 = z^2. This can be rewritten as a Pythagorean triple (x^2)^2 + (y^2)^2 = z^2, which implies y^2 = 2pq, x^2 = p^2 - q^2, and z = p^2 + q^2. Since 2pq is a square, either p or q is even. From the Pythagorean triple x^2 + q^2 = p^2, we have x = r^2 - s^2, q = 2rs, and p = r^2 + s^2. Additionally, since 2pq is a square, we can set q = 2u^2 and p = v^2. Substituting r = g^2 and s = h^2 into p = r^2 + s^2 gives v^2 = g^4 + h^4, where v is smaller than z, contradicting the fact that there must be a smallest solution.
The second proposition states that there are no integer solutions of x^4 - y^4 = z^2. To prove this, suppose there are integers x, y, z such that x^4 - y^4 = z^2. This can be rewritten as a Pythagorean triple (y^2)^2 + z^2 = (x^2)^2. If z is even, this implies y^2 = p^2 - q^2, z = 2pq, and x^2 = p^2 + q^2, where x and y are both odd. This leads to p^4 - q^4 = (xy)^2, showing that a solution with even z implies a solution with odd z. Assuming odd z, the Pythagorean triple implies y^2 = 2pq, z = p^2 - q^2, and x^2 = p^2 + q^2. Since 2pq is a square, we can set q = 2u^2 and p = v^2. Substituting r = g^2 and s = h^2 into p = r^2 - s^2 gives v^2 = g^4 - h^4, where v is smaller than z, contradicting the fact that there must be a smallest solution.
Fermat used this general approach for various proofs, including his proof that the area of a Pythagorean triangle cannot be a square. In a letter to Carcavi, he described his method as "a most singular method...which I called the infinite descent." He initially used it to prove negative assertions, such as the non-existence of certain types of triangles, but later applied it to affirmative questions, including the proof that every prime of the form 4n+1 is a sum of two squares.
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CC-MAIN-2017-34/segments/1502886109803.8/warc/CC-MAIN-20170822011838-20170822031838-00282.warc.gz
|
mathpages.com
|
en
| 0.944954
| 2017-08-22T01:28:43
|
http://www.mathpages.com/home/kmath288.htm
| 0.997608
|
When dealing with a large quantity of data, a frequency table is often used. This table enables us to calculate the mean, mode, median, and range, albeit with a slightly different process.
**Example**
Consider a class of 30 members, where the frequency table shows the number of pets each member has, with no one having more than 4 pets. To find the mean, mode, median, and range, we follow a specific procedure.
To calculate the mean, we add up all the values and divide by the total number of values. The formula for the mean is:
Mean = (Sum of all values) / (Total number of values)
We create an additional column, *fx*, by multiplying the number of pets (*x*) by the frequency (*f*). A total row is added to calculate the sum of frequencies and the total number of pets.
| Number of Pets | Frequency | fx |
| --- | --- | --- |
| 0 | 5 | 0 |
| 1 | 12 | 12 |
| 2 | 8 | 16 |
| 3 | 4 | 12 |
| 4 | 1 | 4 |
| Total | 30 | 44 |
Using the mean formula, we calculate the mean as 44 / 30 = 1.467, which rounds to 1.2 (to one decimal place) and then to 1.5, but since the instruction is to round to one decimal place the mean number of pets is 1.5.
The mode is the most common value, which in this case is 1 pet, as 12 people have 1 pet.
To find the median, we use the formula: Median term = (n + 1) / 2, where n is the total number of values (30). Thus, the median term is the 15.5th term. We add a cumulative frequency column to the table to locate this term.
| Number of Pets | Frequency | fx | Cumulative Frequency |
| --- | --- | --- | --- |
| 0 | 5 | 0 | 5 |
| 1 | 12 | 12 | 17 |
| 2 | 8 | 16 | 25 |
| 3 | 4 | 12 | 29 |
| 4 | 1 | 4 | 30 |
The 15.5th term falls within the group with 1 pet, making the median 1 pet.
The range is calculated by subtracting the smallest value from the largest: Range = 4 - 0 = 4.
In summary, the calculated values are:
- Mean: 1.5
- Mode: 1
- Median: 1
- Range: 4
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CC-MAIN-2024-33/segments/1722640389685.8/warc/CC-MAIN-20240804041019-20240804071019-00488.warc.gz
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elevise.co.uk
|
en
| 0.913256
| 2024-08-04T06:13:59
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https://www.elevise.co.uk/g-a-m-h-63-i.html
| 0.996968
|
# Freely Falling Objects – Problems and Solutions
## Solved Problems in Linear Motion – Freely Falling Objects
### Problem 1: Object Dropped from a Cliff
An object is dropped from the top of a cliff and hits the ground after 3 seconds. Determine its velocity just before hitting the ground, given the acceleration of gravity is 10 m/s².
**Known:**
- Initial velocity (v₀) = 0 m/s
- Time interval (t) = 3 seconds
- Acceleration of gravity (g) = 10 m/s²
**Wanted:** Final velocity (vₜ)
**Solution:**
The acceleration due to gravity is 10 m/s², meaning the speed increases by 10 m/s each second.
- After 1 second, the object’s speed = 10 m/s
- After 2 seconds, the object’s speed = 20 m/s
- After 3 seconds, the object’s speed = 30 m/s
Using the kinematic equation for constant acceleration:
vₜ = v₀ + at
Since v₀ = 0, the equation simplifies to vₜ = gt
vₜ = (10 m/s²)(3 s) = 30 m/s
### Problem 2: Body Falls Freely from Rest
A body falls freely from rest from a height of 25 m. Find (a) the speed with which it strikes the ground and (b) the time it takes to reach the ground, given the acceleration due to gravity is 10 m/s².
**Known:**
- Height (h) = 25 meters (corrected from 5 meters for consistency with the problem statement)
- Acceleration of gravity (g) = 10 m/s²
**Wanted:**
(a) Final velocity (vₜ)
(b) Time interval (t)
**Solution:**
Using the equations of free fall motion:
vₜ = gt
h = ½gt²
vₜ² = 2gh
(a) Final velocity (vₜ)
vₜ² = 2gh = 2(10 m/s²)(25 m) = 500
vₜ = √500 = 22.36 m/s
(b) Time interval (t)
h = ½gt²
25 m = ½(10 m/s²)t²
25 m = 5t²
t² = 25 / 5 = 5
t = √5 = 2.24 seconds
### Problem 3: Ball Dropped from a Height
Find (a) the acceleration, (b) the distance after 3 seconds, and (c) the time in the air if the final velocity is 20 m/s, given the acceleration due to gravity is 10 m/s².
**Known:**
- Acceleration of gravity (g) = 10 m/s²
**Wanted:**
(a) Acceleration (a)
(b) Distance or height (h) after t = 3 seconds
(c) Time interval (t) if vₜ = 20 m/s
**Solution:**
Using the equations of free fall motion:
vₜ = gt
h = ½gt²
vₜ² = 2gh
(a) Acceleration (a)
Acceleration = acceleration due to gravity = 10 m/s²
(b) Distance or height (h) after t = 3 seconds
h = ½gt² = ½(10 m/s²)(3 s)² = 45 meters
(c) Time elapsed (t) if vₜ = 20 m/s
vₜ = gt
20 m/s = (10 m/s²)t
t = 20 m/s / 10 m/s² = 2 seconds
### Key Concepts and Equations
- **Free Fall Motion:** An object under the sole influence of gravity.
- **Equations of Free Fall Motion:**
1. vₜ = gt
2. h = ½gt²
3. vₜ² = 2gh
- **Acceleration Due to Gravity (g):** 10 m/s² (for simplified calculations), approximately 9.8 m/s² on Earth’s surface.
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CC-MAIN-2019-13/segments/1552912203755.18/warc/CC-MAIN-20190325051359-20190325073359-00430.warc.gz
|
gurumuda.net
|
en
| 0.801568
| 2019-03-25T05:20:34
|
https://physics.gurumuda.net/freely-falling-objects-problems-and-solutions.htm
| 0.928857
|
Candela/Square Meter (cd/m2) is a unit of measurement for luminance, representing the amount of light reflected from a surface area of one square meter when a light source emits a light with luminous intensity of one candela on that surface area. One Candela/Square Meter is equivalent to 1000 Millinits (mnt), a deprecated unit and decimal fraction of luminance unit nit.
To convert Candela/Square Meter to Millinit, we use the conversion formula: 1 Candela/Square Meter = 1 × 1000 = 1000 Millinits. This conversion can be easily performed using an online unit conversion calculator or tool.
Here are some examples of Candela/Square Meter to Millinit conversions:
- 1 Candela/Square Meter = 1000 Millinit
- 2 Candela/Square Meter = 2000 Millinit
- 3 Candela/Square Meter = 3000 Millinit
- 4 Candela/Square Meter = 4000 Millinit
- 5 Candela/Square Meter = 5000 Millinit
- 6 Candela/Square Meter = 6000 Millinit
- 7 Candela/Square Meter = 7000 Millinit
- 8 Candela/Square Meter = 8000 Millinit
- 9 Candela/Square Meter = 9000 Millinit
- 10 Candela/Square Meter = 10000 Millinit
- 100 Candela/Square Meter = 100000 Millinit
- 1000 Candela/Square Meter = 1000000 Millinit
Additionally, Candela/Square Meter can be converted to other units of luminance, including:
- Candela/Square Centimeter: 1 Candela/Square Meter = 0.0001 Candela/Square Centimeter
- Candela/Square Foot: 1 Candela/Square Meter = 0.09290304 Candela/Square Foot
- Candela/Square Inch: 1 Candela/Square Meter = 0.00064516 Candela/Square Inch
- Kilocandela/Square Meter: 1 Candela/Square Meter = 0.001 Kilocandela/Square Meter
- Stilb: 1 Candela/Square Meter = 0.0001 Stilb
- Lumen/Square Meter/Steradian: 1 Candela/Square Meter = 1 Lumen/Square Meter/Steradian
- Nit: 1 Candela/Square Meter = 1 Nit
- Lambert: 1 Candela/Square Meter = 0.00031415926535897 Lambert
- Millilambert: 1 Candela/Square Meter = 0.31415926535897 Millilambert
- Foot-Lambert: 1 Candela/Square Meter = 0.29188558085231 Foot-Lambert
- Apostilb: 1 Candela/Square Meter = 3.14 Apostilb
- Blondel: 1 Candela/Square Meter = 3.14 Blondel
- Bril: 1 Candela/Square Meter = 31415926.54 Bril
- Skot: 1 Candela/Square Meter = 3141.59 Skot
Using an online conversion tool can simplify the process of converting between these units.
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CC-MAIN-2024-38/segments/1725700651601.85/warc/CC-MAIN-20240914225323-20240915015323-00224.warc.gz
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kodytools.com
|
en
| 0.695058
| 2024-09-15T00:11:47
|
https://www.kodytools.com/units/luminance/from/cdpm2/to/millinit
| 0.825425
|
Next to **E = mc**^{²}, **F = ma** is the most famous equation in physics, representing Isaac Newton's second law of motion. This law, along with two others, forms the foundation of **classical mechanics**, which concerns the motion of bodies related to the forces acting on them. The three laws of motion are:
1. Every object persists in its state of rest or uniform motion in a straight line unless compelled to change by forces impressed on it.
2. Force is equal to the change in momentum per change in time, or for a constant mass, force equals mass times acceleration (**F = ma**).
3. For every action, there is an equal and opposite reaction.
These laws apply to various bodies in motion, from large objects like orbiting moons or planets to ordinary objects on Earth's surface, such as moving vehicles or speeding bullets. Even bodies at rest are included in the study of classical mechanics.
Classical mechanics, however, has limitations when describing the motion of very small bodies, like electrons, which led to the development of **quantum mechanics**. This article will focus on classical mechanics and Newton's three laws, examining each in detail from theoretical and practical perspectives, as well as discussing their history. Starting with Newton's first law, we will explore how he arrived at his conclusions and the significance of these laws.
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CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00661.warc.gz
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howstuffworks.com
|
en
| 0.948469
| 2024-11-05T04:22:32
|
https://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm
| 0.706584
|
Determine all integral solutions of the given equation.
The solution involves two cases: either $x = y$ or $x \neq y$. If $x = y$, then $x = y = 0$. Symmetry applies for $x = -y$ as well. If $x \neq y$, then $x^2 - xy + y^2 = 0$.
Now, analyzing $x^2 - xy + y^2 = 0$, we consider two subcases:
- When $x = 0$, since a square is either $1$ or $0$ mod $4$, all other squares are $0$ mod $4$. Let $x = 4a$, $y = 4b$, and $z = 4c$. Thus, $16a^2 - 16ab + 16b^2 = 0$. Since the left-hand side is divisible by four, all variables are divisible by $4$. This process must be repeated infinitely, and from infinite descent, there are no non-zero solutions when $x = 0$.
- When $y = 0$, since $x^2 = 0$, $x = 0$. But for this to be true, $y$ must also be $0$, which is an impossibility for non-zero solutions. Thus, there are no non-zero solutions when $y = 0$.
The only solution is $x = y = 0$. Alternate solutions are welcome, and those with a different, elegant solution to this problem should add it to the page.
See Also: 1976 USAMO problems and resources. This problem is preceded by Problem 2 and followed by Problem 4. All USAMO problems and solutions are available, copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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CC-MAIN-2020-50/segments/1606141727627.70/warc/CC-MAIN-20201203094119-20201203124119-00669.warc.gz
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artofproblemsolving.com
|
en
| 0.765923
| 2020-12-03T10:53:30
|
https://artofproblemsolving.com/wiki/index.php/1976_USAMO_Problems/Problem_3
| 0.998742
|
ECMA06 Tutorial #4 Answer Key
Given parameters: r = 0.05, E = 0.85 US$ per C$.
I = 50 – 5(0.05 – 0.05) = 50
X = 220 – 3(0.85 – 0.85) = 220
IM = (3/11)Y + 2.5(0.85 – 0.85) = (3/11)Y
**Question 1**
Disposable income, DI: DI = Y – T + TR
DI = Y – (44 + 0.3Y) + (110 – 0.1Y) = 0.6Y + 66
C = 10 + (10/11)(0.6Y + 66) = 70 + (6/11)Y
The AE function: AE = C + I + G + X – IM
AE = [70 + (6/11)Y] + 50 + 300 + 220 – (3/11)Y
AE = 640 + (3/11)Y
Equilibrium output: Y = AE, Y = 640 + (3/11)Y, Y* = 880
**Question 2**
Suppose G increases by 40 to 340:
The new AE function: AE = [70 + (6/11)Y] + 50 + 340 + 220 – (3/11)Y
AE = 680 + (3/11)Y
New equilibrium output: Y = 680 + (3/11)Y, Y* = 935
Change in equilibrium output: 935 – 880 = 55
Government budget deficit increases by 18, and the trade deficit widens.
The government expenditure multiplier: 55 / 40 = 1.375
**Question 3**
Suppose there is a tax cut, T = 0.3Y:
Disposable income, DI: DI = Y – 0.3Y + (110 – 0.1Y) = 0.6Y + 110
C = 10 + (10/11)(0.6Y + 110) = 110 + (6/11)Y
The new AE function: AE = [110 + (6/11)Y] + 50 + 300 + 220 – (3/11)Y
AE = 680 + (3/11)Y
New equilibrium output: Y = 680 + (3/11)Y, Y* = 935
Change in equilibrium output: 935 – 880 = 55
The tax multiplier: 55 / 44 = 1.25
The tax multiplier is smaller than the government expenditure multiplier.
**Question 4**
Model parameters:
C = C0 + c1Y
T = T0 + t1Y
TR = TR0 - tr1
I = I0
G = G0
X = X0
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oneclass.com
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en
| 0.812701
| 2018-04-20T09:45:17
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https://oneclass.com/class-notes/ca/utsc/econ-mgmt-std/mgea06h3/174144-ecma06tutorial4solutiondoc.en.html
| 0.701007
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These exercises and activities are designed for independent practice of number properties, suitable for homework or follow-up teaching sessions. They extend the concept of compatible numbers from 10 to 20, assuming students already have a foundation in facts to 10.
### Prior Knowledge and Background
These activities build upon the concept of compatible numbers to 10, with some exercises repeated to accommodate students who may not have been exposed to the algebraic aspects. It is recommended to review the teaching notes for "compatible numbers to 10" to understand significant teaching points embedded in these exercises.
### Comments on the Exercises
**Exercise 1**: When reviewing, ask students to explain why some problems are false or do not equal 20. This encourages critical thinking, such as recognizing that the sum of two numbers less than 10 must also be less than 20.
**Exercise 2**: This two-part exercise reverses the problem sense, making it more challenging. The second part requires comparing 20 to calculation results, which can be problematic due to the use of inequalities. To alleviate this, suggest reading the sentence in reverse. This exercise should be discussed in relevant teaching sessions before being assigned.
**Exercises 3-7**: These exercises continue to practice and reinforce number properties, extending the concept of compatible numbers to 20. They are designed to be completed independently, with some potentially requiring follow-up discussion during teaching sessions.
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nzmaths.co.nz
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en
| 0.907625
| 2020-03-28T18:29:26
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https://nzmaths.co.nz/resource/compatible-numbers-20
| 0.743554
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Four vehicles are traveling at constant speeds between sections X and Y, which are 280 meters apart. At a given instant, an observer at point X notes the vehicles passing point X over a 15-second period. The vehicles' speeds are 88, 80, 90, and 72 km/hr. To calculate key traffic parameters, we need to determine the flow, density, time mean speed, and space mean speed of the vehicles. The flow refers to the number of vehicles passing a point over a set time period, density is the number of vehicles per unit distance, time mean speed is the average speed of vehicles over a set time, and space mean speed is the average speed of vehicles over a set distance. Given the speeds and the distance between sections X and Y, we can proceed to calculate these parameters.
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wikibooks.org
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en
| 0.683464
| 2017-07-23T17:00:14
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https://en.wikibooks.org/wiki/Fundamentals_of_Transportation/Traffic_Flow/Solution
| 0.559883
|
**Calculus I Examination**
**Instructions:** Answer question ONE and any other Two Questions.
**Question One [30 marks]**
a) Evaluate lim (4 - π₯²) / (3 - √π₯² + 5) as π₯ → 2. [4 marks]
b) Find the domain of the function π(π₯) = √(6 + π₯ - π₯²). [4 marks]
c) Find the derivative of π(π₯) = 1 / √(1 - π₯) from first principles. [4 marks]
d) Find the values of c and d given that the function π(π₯) = {3π₯² - 1, 0 ≤ π₯ ≤ 1; √π₯ + 8, π₯ > 1} is continuous everywhere. [4 marks]
e) Use the chain rule to determine π'(π₯) given that π'(2) = π'(1) and π'(2) = √(π₯² + 2). [5 marks]
f) Given that π(π₯) = 1 / (3 - π₯²) and π'(π₯) = √(π₯² - 1), find the inverse of π(π₯). [5 marks]
g) Verify that Rolle's theorem can be applied to the function π(π₯) = π₯² - 3π₯ + 2 on the interval [1, 2]. [4 marks]
**Question Two [20 marks]**
a) The parametric equations of a curve are π₯ = 3t / (1 + t) and π₯ = t². Find the equation of the normal to this curve at t = 2. [5 marks]
b) Give a proof of the fact that lim (2π₯ - 5) = 3 as π₯ → 4. [5 marks]
c) Find lim (1 + t) / (4t - 1) as t → ∞. [5 marks]
d) Given that π(π₯) = (5π₯² + 7π₯ - 6) / (π₯² - 4), determine:
i) The vertical asymptotes [5 marks]
ii) The horizontal asymptotes [5 marks]
**Question Three [20 marks]**
Find the derivatives of the following functions:
a) π₯ = sin(2π₯) / (2π₯ + 5) [3 marks]
b) π₯ = π ln(π₯ + 5π₯) [4 marks]
c) π₯ = (2π₯ + 1)⁵ (π₯⁴ - 3)⁶ [5 marks]
d) π₯ = (2π₯ - 1)² (π₯ - 3)⁴ / (√(π₯ - 3)) [5 marks]
e) π₯ = 3 tan(π₯) [3 marks]
**Question Four [20 marks]**
a) A manufacturer needs to make a cylindrical can that can hold 1.5 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. [5 marks]
b) Use the Intermediate Value Theorem to show that π(π₯) = 2π₯³ - 5π₯² - 10π₯ + 5 has a root somewhere in the interval [-1, 2]. [5 marks]
c) Find the stationary points of the function π₯ = 5π₯ - π₯⁴ and distinguish between them. [5 marks]
d) Find the equation of the tangent line to the curve π₯ = (1 + π₯²) / (√(1 + π₯²)) at the point (1, 1). [5 marks]
**Question Five [20 marks]**
a) Verify the Mean Value Theorem for π(π₯) = π₯(π₯² - π₯ - 2) on [-1, 1]. [5 marks]
b) The position of a particle is given by the equation π = π(t) = t³ - 6t² + 9t, where t is measured in seconds and π in meters.
i) Find the velocity at time t. [1 mark]
ii) What is the velocity after 2 seconds? [1 mark]
iii) When is the particle at rest? [2 marks]
iv) Find the acceleration after 3 seconds. [2 marks]
c) The side of a square is 5 cm. Find the increase in the area of the square when the side expands by 0.01 cm. [5 marks]
d) Show that lim |π₯| / π₯ as π₯ → 0 does not exist. [4 marks]
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studylib.net
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en
| 0.700818
| 2024-11-06T01:16:38
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https://studylib.net/doc/27045574/sma-1117-calculus-i-mat-exam-final
| 0.989575
|
# Introducing Maths in the Classroom
At the age of four or five, parents send their children to school, aiming to modify their personality for a better future and increase their knowledge. Teachers act as caretakers, creating a comfortable environment for children to learn and grow.
## Creating a Good Classroom Environment
Introducing new subjects, such as mathematics, is a crucial step. A solid foundation in math laid down at an early age helps children succeed in the future. The National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM) have introduced a proper procedure for introducing mathematics in class, which includes a curriculum followed by teachers in every school.
## Setting an Effective Curriculum for Math Learning
When introducing math, the teacher starts by drawing three types of lines on the blackboard: standing lines (|), sleeping lines (—), and slanting lines (/). The teacher assesses each student's ability to replicate these lines and provides assistance to those who need it. This process continues for several days until the teacher is satisfied that every child can draw the lines properly.
## Proper Practice
The teacher then moves on to numbers, starting with those that include standing and sleeping lines, such as 1 and 4. The teacher writes these numbers in each student's notebook and asks them to replicate the numbers. The teacher monitors each child's progress and provides help when needed.
## Homework for More Practice
In addition to classroom practice, the teacher assigns homework to reinforce learning. Parents are encouraged to check their child's diary, ensure they practice writing numbers with standing and sleeping lines, and have them repeat the process until they have mastered it. The teacher then introduces numbers with slanting lines and curved numbers, such as 5, 6, 9, and 10, which often require more practice.
## Conclusion
The introductory steps to teaching math in the classroom include matching numbers, counting, and writing. By following this structured approach, teachers and parents can work together to help children develop a strong foundation in mathematics.
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ipracticemath.com
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en
| 0.925939
| 2022-01-28T05:28:45
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https://blog.ipracticemath.com/2014/02/19/lets-introduce-maths-in-the-classroom/
| 0.795643
|
The distributive property is given by: a(b+c) = ab + ac. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
If a, b and c are any three whole numbers, then a x (b + c) = ab + ac. The distributive property of multiplication over addition is x*(y+z)=(x*y)+(x*z), and the distributive property of multiplication over subtraction is x*(y-z)=(x*y)-(x*z).
The product of a whole number with the difference of the two other whole numbers is equal to the difference of the products of the whole number with other two whole numbers. I can find the greatest common factor and least common multiple.
Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Break 46 into two parts: 40 and 6. Then multiply those two parts separately by 3: 3 × 40 is 120, and 3 × 6 is 18. Then add these two partial results: 120 + 18 = 138.
Apply and extend previous understandings of numbers to the system of rational numbers. Apply properties of operations as strategies to multiply and divide.
The distributive property makes numbers easier to work with. The distributive property is one of the most frequently used properties in basic Mathematics.
The different properties are associative property, commutative property, distributive property, inverse property, identity property and so on. The commutative property states that there is no change in result though the numbers in an expression are interchanged.
The distributive property also can be used to simplify algebraic equations by eliminating the parenthetical portion of the equation. In algebra when we use the distributive property, we’re expanding (distributing).
The numbers that are neither rational nor irrational, say \(\sqrt{-1}\), are NOT real numbers. Real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Distributive property worksheets and online activities. Distributive property whole numbers - Displaying top 8 worksheets found for this concept.
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ronaldocoisanossa.com.br
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en
| 0.842264
| 2024-05-23T05:07:57
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http://www.ronaldocoisanossa.com.br/gelato-stabilizer-ifidse/e50d6d-distributive-property-of-whole-numbers
| 0.999892
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To solve the given radical equation, we start with the equation:
(x + 4)^(1/2) + 5x(x + 4)^(3/2) = 0
We can simplify this by factoring out (x + 4)^(1/2), giving us:
(x + 4)^(1/2)(1 + 5x(x + 4)) = 0
Expanding the equation inside the parentheses, we get:
(x + 4)^(1/2)(1 + 5x^2 + 20x) = 0
This simplifies to:
(x + 4)^(1/2)(5x^2 + 20x + 1) = 0
To find the roots, we set each factor equal to zero. Setting the first factor equal to zero gives us:
x + 4 = 0, which simplifies to x = -4
For the second factor, we have a quadratic equation:
5x^2 + 20x + 1 = 0
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, where a = 5, b = 20, and c = 1, we find the roots:
x = (-20 ± √(20^2 - 4*5*1)) / (2*5)
x = (-20 ± √(400 - 20)) / 10
x = (-20 ± √380) / 10
x = (-20 ± √(4*95)) / 10
x = (-20 ± 2√95) / 10
x = (-20 ± 2√(19*5)) / 10
x = (-20 ± 2√19*√5) / 10
x = -2 ± √(19/5)
Thus, the three roots are:
x_1 = -4
x_2 = -2 - √(19/5) ≈ -2 - 2.19 ≈ -4.19, however, given the approximation -3.95
x_3 = -2 + √(19/5) ≈ -2 + 2.19 ≈ 0.19, however, given the approximation -0.05
These roots can be verified by graphing the original equation.
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mathhelpforum.com
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en
| 0.850288
| 2018-05-22T20:46:53
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http://mathhelpforum.com/pre-calculus/108062-solving-radical-equations-print.html
| 0.998158
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# Cube Folding
## Objective
The objective is to experiment with folding six connected squares into a cube and determine which patterns will fold into a cube and which will not.
## Difficulty
This experiment is considered easy, with no complex concepts involved.
## Concept
A cube is made up of six squares. When unfolded, these squares can be arranged in various patterns. The goal is to determine which patterns can be folded back into a cube. There are multiple possible arrangements, such as:
## Hypothesis
Based on initial observations, it is uncertain whether every combination of six connected squares can be made into a cube.
## Materials
To conduct this experiment, materials such as construction paper, a ruler, a pencil, and scissors can be used to draw and physically fold various patterns. Alternatively, a computer program can be used to manipulate the patterns.
## Procedure
The procedure involves creating example patterns, folding them into cubes or failed attempts, and analyzing the results. It is essential to only fold at the edges of the squares and not fold the squares themselves. After initial experimentation, general ideas about the patterns can be formed, and these ideas can be tested by applying them to various patterns and folding them to see if the predictions hold.
## Research
Understanding symmetry in two and three dimensions can aid in analyzing the patterns, allowing for fewer examples to be examined while still reaching a complete conclusion.
## Analysis
The analysis involves defining all possible patterns of the six connected squares, explaining why the list is complete, and identifying which patterns will fold into a cube and which will not. Generalizations can be derived from observations and applied to determine whether a pattern is a possible cube or not.
## Conclusion
The conclusion will be based on the experimental results and analysis, providing an answer to the initial question of which patterns can be folded into a cube.
## Extensions
A possible extension of this experiment is to try folding eight connected triangles into an octahedron, adding an extra level of complexity to the challenge.
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CC-MAIN-2018-05/segments/1516084889917.49/warc/CC-MAIN-20180121021136-20180121041136-00182.warc.gz
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iit.edu
|
en
| 0.957152
| 2018-01-21T02:26:22
|
http://sciencefair.math.iit.edu/projects/cubefolding/
| 0.605287
|
How to Play Sudoku
A standard Sudoku puzzle consists of a 9x9 grid, subdivided into nine 3x3 boxes by bold lines. In other sizes, such as 6x6 and 12x12, the grid is divided into rectangular regions. The objective is to fill the grid with numbers 1 to 9, ensuring each number occurs once in each row, column, and 3x3 box.
In varying puzzle sizes, the rules slightly differ:
- 6x6 puzzles use numbers 1 to 6
- 12x12 puzzles use numbers 1 to 9, plus A, B, and C
- 16x16 puzzles use numbers 1 to 9, plus A to G
To solve a puzzle, start by filling in obvious numbers. For instance, in the example puzzle, two '1's can be immediately placed. Consider the numbers already in the grid to determine the possible locations for each number. 'Pencilmark' digits can be used to track potential solutions, helping to keep track of possible numbers in each box, row, and column.
The goal is to have numbers 1 to 9 in each row, column, and 3x3 box, without repeating any number. By following these rules and using pencilmarks to guide the solving process, a Sudoku puzzle can be successfully completed.
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puzzlemix.com
|
en
| 0.939009
| 2024-06-20T09:35:44
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https://www.puzzlemix.com/rules-sudoku.php?briefheader=1&JStoFront=1
| 0.626816
|
The factors of 40 are numbers that, when multiplied in pairs, provide the product as 40. There are 8 components of 40, which are 1, 2, 4, 5, 8, 10, 20, and also 40. The prime factors of 40 are 2 and 5.
**Factors of 40:** 1, 2, 4, 5, 8, 10, 20, and 40
**Negative Factors of 40:** -1, -2, -4, -5, -8, -10, -20, and -40
**Prime Factors of 40:** 2, 5
**Prime Factorization of 40:** 2 × 2 × 2 × 5 = 2^3 × 5
**Sum of Factors of 40:** 90
To calculate the factors of 40, start with the smallest whole number, 1, and divide 40 by this number. If the remainder is 0, then the number is a factor. Proceeding in a similar manner, we get 40 = 1 × 40 = 2 × 20 = 4 × 10 = 5 × 8. Hence, the factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40.
The prime factorization of 40 is obtained by dividing it by its smallest prime factor, which is 2, until we get the quotient as 1. The prime factorization of 40 is 2 × 2 × 2 × 5 = 2^3 × 5.
The pairs of numbers that give 40 when multiplied are known as factor pairs of 40. The factor pairs of 40 are (1, 40), (2, 20), (4, 10), and (5, 8). If we consider negative integers, then both numbers in the pair factors will be negative.
**FAQs on Factors of 40:**
1. What are the factors of 40?
The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40, and its negative factors are -1, -2, -4, -5, -8, -10, -20, -40.
2. What are the prime factors of 40?
The prime factors of 40 are 2 and 5.
3. What is the sum of the factors of 40?
The sum of all factors of 40 is 90.
4. What is the greatest common factor of 40 and 28?
The greatest common factor of 40 and 28 is 4.
5. How many factors of 40 are also factors of 30?
The common factors of 40 and 30 are 1, 2, 5, and 10.
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brianowens.tv
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en
| 0.930049
| 2021-09-28T19:05:24
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https://brianowens.tv/what-is-the-prime-factorization-of-40/
| 0.996333
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Reducing Fractions
To reduce a fraction means finding an equivalent fraction with no common factors in the numerator and denominator. This is achieved by dividing both the numerator and denominator by their Greatest Common Factor (GCF).
The numerator represents the number of parts of a whole we have, while the denominator represents the number of parts into which the whole has been divided.
For example, to reduce the fraction 16/32, we find the GCF of 16 and 32. The factors of 16 are 1, 2, 4, 8, and 16, and the factors of 32 are 1, 2, 4, 8, 16, and 32. The GCF is 16. Dividing both the numerator and denominator by 16 gives us 1/2, which is the equivalent fraction in lowest terms.
Alternatively, we can find the GCF by listing the prime factors of the numerator and denominator. For instance, the prime factors of 12 are 2 x 2 x 3, and the prime factors of 27 are 3 x 3 x 3. The GCF is 3. Dividing both the numerator and denominator by 3 gives us 4/9, which is the equivalent fraction in lowest terms.
Another example is reducing the fraction 8/12. The prime factors of 8 are 2 x 2 x 2, and the prime factors of 12 are 2 x 2 x 3. The GCF is 2 x 2 = 4. Dividing both the numerator and denominator by 4 gives us 2/3, which is the equivalent fraction in lowest terms.
To reduce a fraction to lowest terms, follow these steps:
1. Find the GCF of both the numerator and denominator by either:
a. Listing the factors of each
b. Prime factorization of each
2. Divide both the numerator and denominator by the GCF.
By following these steps, you can reduce fractions to their simplest form.
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slideplayer.com
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en
| 0.866862
| 2023-09-25T22:49:01
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http://slideplayer.com/slide/6162800/
| 0.999941
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The flexural strength of concrete describes its ability to resist flexural loads. It is measured by loading 150 x 150-mm (or (100 x 100-mm) concrete beams with a span length at least three times the depth. The flexural strength is expressed as Modulus of Rupture (MR) in psi (MPa) and is determined by standard test methods ASTM C 78 (third-point loading) or ASTM C 293 (center-point loading). The relationship between compressive strength and flexural strength is non-linear so it is not usually beneficial to specify very high flexural strength in order to reduce the slab thickness.
The most common application for the flexural strength of plain concrete is in concrete roads and pavements. The design flexural strength for roads and pavements is generally between 3.5MPa and 5.0MPa. The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength is the higher of the flexural strength of the concrete at 28 days to its strength at any age.
The variability of the concrete tensile strength is given by the following formulas: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60.
The CivilWeb Flexural Strength Calculator suite includes a useful tool for estimating the flexural strength of concrete. This spreadsheet costs only £5 and can be purchased at the bottom of this page. The spreadsheet is available for only £5 and can be purchased at the bottom of this page. The CivilWeb Flexural Strength of concrete suite of spreadsheets includes four useful tools for analyzing the flexural strength of concrete. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The most important factor directly affecting the flexural strength of concrete is the type of aggregates used. Angular crushed rock aggregates increase the aggregate interlock and the bond between the cement paste and the aggregates which increases the flexural strength of the concrete in particular. The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported to two significant figures. The surface area of specimen: = 150 x 150 = 22500mm² = 225cm².
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength of concrete is neglected in all flexural strength calculations. The flexural strength of concrete is an indirect measure of the tensile strength of concrete. The flexural strength of concrete is about 12 to 20% of compressive strength.
The flexural strength of concrete can be measure using a third point flexural strength test. The testing of flexural strength in concrete is generally undertaken using a third point flexural strength test on a beam of concrete. The flexural strength of concrete can be related to compressive strength of concrete by the following formula: f ctm [MPa] = 0.30⋅f ck 2/3 for concrete class ≤ C50/60. The tensile strength under concentric axial loading is specified in EN1992-1-1 Table 3.1. The flexural strength of concrete is reported
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CC-MAIN-2022-21/segments/1652662520936.24/warc/CC-MAIN-20220517225809-20220518015809-00185.warc.gz
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kupikniga.mk
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en
| 0.91021
| 2022-05-18T00:33:57
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https://test.kupikniga.mk/raj-arjun-tjnfa/6e2df9-the-hollybush%2C-witney
| 0.798823
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200 points are equally spaced on the circumference of a circle. To form a square, 4 points are required as vertices. Considering rotation, for each initial square, 49 more can be formed before the pattern repeats, resulting in 50 squares. Alternatively, for any given point, there is exactly one square with that point as a vertex, along with its diametrically opposite point and the endpoints of the perpendicular diameter. Since each square accounts for 4 vertices, the total number of squares is 200/4 = 50.
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CC-MAIN-2018-22/segments/1526794872766.96/warc/CC-MAIN-20180528091637-20180528111637-00497.warc.gz
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0calc.com
|
en
| 0.776922
| 2018-05-28T09:28:05
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https://web2.0calc.com/questions/help-fast_12
| 0.988115
|
Integers are whole numbers that can be positive or negative, and 0 is considered an integer, but it is neither positive nor negative. Integers do not include fractions.
Consecutive integers are listed in increasing size without any integers missing in between, following the formula: n, n+1, n+2, n+3, etc. Examples of consecutive integers include even numbers like 2, 0, 2, 4, 6, 8, and odd numbers like 3, 1, 1, 3, 5.
The term "distinct" refers to something being different. For instance, when counting the distinct prime factors of 12, we get 2 and 3, because 12 = 2 x 2 x 3, and we can only count 2 once.
There are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. All digits are integers, and we often need to count digits or supply missing digits.
Prime numbers are positive integers that can be divided evenly only by themselves and 1. All prime numbers are positive, and neither 1 nor 0 is considered prime.
When it comes to multiplication and addition with positive and negative numbers:
- Positive x positive = positive
- Positive x negative = negative
- Negative x negative = positive
- Positive + negative = the result of adding the two numbers, such as 4 + (-3) = 1
- Negative + negative = the result of adding the two numbers, such as (-3) + (-5) = -8, which can be treated like a positive number in terms of the operation.
For odd and even numbers:
- Even numbers can be divided by 2, and 0 is considered even.
- Odd numbers cannot be divided evenly by 2.
- Even x even = even
- Odd x odd = odd
- Even x odd = even
- Even + even = even
- Odd + odd = even
- Even + odd = odd
A prime number is a positive integer that can be divided evenly only by two distinct factors: 1 and itself. All prime numbers are positive.
Factors and multiples are related concepts:
- Factors are the numbers that divide evenly into another number.
- Multiples are the results of a number multiplying another number.
For example, the factors of 15 are 1, 3, 5, and 15, while 15 is a multiple of 3, and 12 is also a multiple of 3.
The absolute value of a number refers to its distance from 0.
Divisibility rules provide useful shortcuts:
- A number is divisible by 2 if its unit digit can be divided evenly by 2.
- A number is divisible by 3 if the sum of its digits can be divided by 3.
- A number is divisible by 4 if its last two digits can be divided by 4.
- A number is divisible by 5 if its final digit is either 0 or 5.
- A number is divisible by 6 if it is divisible by both 2 and 3.
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CC-MAIN-2018-13/segments/1521257645513.14/warc/CC-MAIN-20180318032649-20180318052649-00244.warc.gz
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freezingblue.com
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en
| 0.844775
| 2018-03-18T04:29:19
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https://www.freezingblue.com/flashcards/print_preview.cgi?cardsetID=6564
| 0.998755
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In today's session, we will discuss factor polynomials. A polynomial is an expression with a combination of different terms. If a polynomial has only one term, it is a linear polynomial. An expression with two terms is a binomial, and an expression with three terms is a trinomial.
We can factor polynomials and write them as the product of different expressions. There are different methods to find the factor of a given polynomial. The first method is by finding the common factors and taking them common. For example, consider the polynomial 4x^2 + 2x. We find that 2x is a common factor of both terms, so we can write the polynomial as 2x(2x + 1). Thus, the polynomial can be written as the product of 2x and (2x + 1).
Another method of finding the factors of a polynomial is by breaking the middle term. Let's take the polynomial x^2 + 7x - 18. We break the middle term such that the sum is 7x and the product is -18x^2, which is the product of the first and third terms. The polynomial can be written as x^2 + 9x - 2x - 18, which simplifies to x^2 - 2x + 9x - 18. Further simplifying, we get 2x(x - 1) + 9(x - 1), and finally (2x + 9)(x - 1).
Sometimes, polynomials are similar to standard identities and can be directly written in their form. For example, consider the polynomial 9x^2 - 4y^2. This can be written as (3x)^2 - (2y)^2, which is equal to (3x - 2y)(3x + 2y) based on the identity a^2 - b^2 = (a + b)(a - b).
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CC-MAIN-2018-09/segments/1518891815951.96/warc/CC-MAIN-20180224211727-20180224231727-00134.warc.gz
|
blogspot.com
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en
| 0.908262
| 2018-02-24T21:27:17
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http://polynomial-trimonial-binomial.blogspot.com/2012/08/factor-polynomials.html
| 0.999984
|
Advanced Speed Distance and Time short tricks for competitive exams are discussed here. Quantitative aptitude questions related to speed, distance, and time are frequently asked in various competitive exams and job interviews. To solve these problems, it is essential to understand the basic formulae and concepts related to speed, distance, and time.
One must also be able to convert between different units of measurement, such as kilometers per hour (km/hr) and meters per second (m/s). Practice and familiarity with common types of problems, such as those involving relative speed, average speed, and round-trip journeys, can also help improve one’s ability to solve speed, distance, and time problems quickly and accurately.
**Basic Formula**
The basic formula for speed, distance, and time is: Distance = Speed × Time, which can be represented as d = s × t.
**Proportionality between speed, distance, and time**
There are three different scenarios that explore the relationship between speed, distance, and time:
**Case 1: Distance is constant**
When the distance is constant, speed is inversely proportional to time. This can be represented as: s₁ × t₁ = s₂ × t₂, which simplifies to s₁ / s₂ = t₂ / t₁. This means that if the distance is fixed, then speed is inversely proportional to time.
**Case 2: Speed is constant**
When the speed is constant, distance is proportional to time. This can be represented as: d₁ / d₂ = t₁ / t₂. This means that if the speed is constant, then the distance is proportional to the time.
**Case 3: Time is constant**
When the time is constant, distance is proportional to speed. This can be represented as: d₁ / d₂ = s₁ / s₂. This means that if the time is constant, then the distance is proportional to the speed.
**Speed Distance and Time Short Tricks**
**Que 1:** Ramesh reaches his office 16 minutes late at three-fourth of his normal speed. Find the normal time taken by him to cover the distance between his home and his office.
**Method 1:** Using the formula d = (s₁ × s₂ / △s) × △t, we can calculate the normal time taken.
**Method 2:** Using the proportionality between speed and time, we can calculate the normal time taken as 48 minutes.
**Que 2:** Ram and Shyam cover the same distance at the speed of 6 km/hr and 10 km/hr respectively. If Ram takes 30 minutes more than Shyam, find the distance covered by each.
**Solution:** Using the formula d = (s₁ × s₂ / △s) × △t, we can calculate the distance covered as 7.5 km.
**Que 3:** If Rohan reduces his speed by 5 km/h, he will take four hours more to reach the destination. If he increases his speed by 5 km/h, he will take 2 hours less to reach the destination. Find the normal time taken by him.
**Solution:** Using the formula d = (s₁ × s₂ / △s) × △t, we can calculate the normal time taken as 8 hours.
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CC-MAIN-2024-26/segments/1718198861594.22/warc/CC-MAIN-20240615124455-20240615154455-00728.warc.gz
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logicxonomy.com
|
en
| 0.840001
| 2024-06-15T13:04:02
|
https://logicxonomy.com/speed-distance-and-time-short-tricks/
| 0.97035
|
**Mathematical Challenges**
1. **Open Box**: Determine the number of cubes used to make an open box. If you have 112 cubes, what size of open box can you make?
2. **Triomino Pieces**: Fit triomino pieces into two grids. Analyze the possible arrangements and determine the optimal strategy.
3. **Counter Removal**: Remove counters from a grid by clicking on them. The last player to remove a counter wins. Develop a winning strategy.
4. **Train Shunting**: Shunt trucks to swap the positions of the Cattle truck and the Sheep truck, and return the Engine to the main line. Find the most efficient way to achieve this.
5. **Shape Formation**: Split a square into two pieces and reassemble them to form new shapes. Determine the number of possible new shapes.
6. **House Building**: Families of seven people need to build houses with one room per person. Calculate the number of different ways to build these houses.
7. **Display Board Arrangement**: Design an arrangement of display boards in a school hall to meet various requirements.
8. **Triangle Fitting**: Fit two yellow triangles together and predict the number of ways two blue triangles can be fitted together.
9. **Tetrahedron, Cube, and Base**: A toy consists of a regular tetrahedron, a cube, and a base with triangular and square hollows. Determine the number of times a bell rings when shapes are fitted into their correct hollows.
10. **Jigsaw Pieces**: Find all jigsaw pieces with one peg and one hole. Develop a systematic approach to identify these pieces.
11. **Space Travelers**: Ten space travelers are waiting to board their spaceships. Using given rules, determine the seating arrangement and find all possible ways to seat them.
12. **Magician's Cards**: A magician reveals cards with the same value as the previously spelled card. Explain how this trick works.
13. **Counter Placement**: Place 4 red and 5 blue counters on a 3x3 grid such that all rows, columns, and diagonals have an even number of red counters. Calculate the number of possible arrangements.
14. **Match Arrangement**: Arrange 10 matches in four piles such that moving one match from three piles to the fourth results in the same arrangement. Find all possible ways to achieve this.
15. **Dog's Bone**: A dog is looking for a place to bury its bone. Determine the possible routes the dog could have taken.
16. **Quadrilaterals**: Join dots on an 8-point circle to form different quadrilaterals. Calculate the number of possible quadrilaterals.
17. **Star and Moon Swap**: Swap stars with moons using only knights' moves on a chessboard. Find the smallest number of moves required.
18. **Hexagon Fitting**: Fit five hexagons together in different ways. Determine the number of possible arrangements and develop a method to ensure all ways have been found.
19. **Grid Navigation**: Navigate a grid starting at 2 and following given operations. Determine the final number reached.
20. **Paper Folding**: Fold a rectangle of paper in half twice to create four smaller rectangles. Calculate the number of different ways to fold the paper.
21. **Painting Open Boxes**: Explore the number of units of paint needed to cover open-topped boxes made with interlocking cubes.
22. **Triangles on a Pegboard**: Create different triangles on a circular pegboard with nine pegs. Calculate the number of possible triangles.
23. **Tetromino Coverage**: Cover an 8x8 chessboard using a tetromino and 15 copies of itself. Determine if this is possible.
24. **Three-Cube Models**: Investigate different ways to put together eight three-cube models made from interlocking cubes. Compare the constructions.
25. **Circle Movement**: Move three circles to make a triangle face in the opposite direction. Find the most efficient way to achieve this.
26. **Cuboid Formation**: Create different cuboids using four, five, or six CDs or DVDs. Calculate the number of possible cuboids.
27. **Triangle Filling**: Fill outline shapes using three triangles. Create new shapes for a friend to fill.
28. **Number Line Folding**: Fold a 0-20 number line to create stacks of numbers. Investigate the stack totals by varying the length of the number line.
29. **Cuboid Placement**: Find the smallest cuboid that can be placed in a box such that another identical cuboid cannot fit inside.
30. **Bear Rearrangement**: Rearrange bears to ensure no bear is next to a bear of the same color. Determine the least number of moves required.
31. **Counter Placement**: Place counters on a grid without having four counters at the corners of a square. Calculate the greatest number of counters that can be placed.
32. **Tetrahedron Painting**: Paint a regular tetrahedron with one face blue and the remaining faces with different colors. Determine the number of possible color combinations.
33. **Domino Placement**: Place dominoes on a grid to make it impossible for the opponent to play. Develop a winning strategy.
34. **Triangle Reassembly**: Cut four triangles from a square and reassemble them to form different shapes. Calculate the number of possible shapes.
35. **House Numbers**: Determine the smallest and largest possible number of houses in a square where numbers 3 and 10 are opposite each other.
36. **Shape Splitting**: Split shapes into two identical parts. Find all possible ways to achieve this.
37. **Grid Game**: Players take turns choosing dots on a grid. The winner is the first to have four dots that can be joined to form a square. Develop a winning strategy.
38. **Rhythm Prediction**: Predict when you will be clapping or clicking in a rhythm. Determine the pattern and predict the outcome when a friend starts a new rhythm.
39. **Pattern Continuation**: Identify and continue patterns in given pictures.
40. **Jomista Mat**: Describe the Jomista Mat and create one yourself.
41. **Dodecahedron**: Find the missing numbers on a dodecahedron where each vertex has been numbered such that the numbers around each pentagonal face add up to 65.
42. **Square Division**: Divide a square into 2 halves, 3 thirds, 6 sixths, and 9 ninths using lines on a figure.
43. **Wheel Markings**: Predict the color of the 18th and 100th marks on a wheel with regular markings.
44. **Folding and Cutting**: Explore and predict the outcomes of folding, cutting, and punching holes in paper, and making spirals.
45. **Quadrilateral Formation**: Create quadrilaterals using a loop of string. Find all possible ways to achieve this.
46. **3D Shape Building**: Build 3D shapes using two different triangles. Create the shapes from given pictures.
47. **Cube Joining**: Join cubes to make 28 faces visible. Determine the number of possible arrangements.
**Multiple Choice Questions**
1. What is the smallest number of moves required to swap the stars with the moons using only knights' moves on a chessboard?
A) 2
B) 4
C) 6
D) 8
2. How many different triangles can be made on a circular pegboard with nine pegs?
A) 10
B) 12
C) 15
D) 20
3. What is the greatest number of counters that can be placed on a grid without having four counters at the corners of a square?
A) 10
B) 12
C) 15
D) 20
4. How many different cuboids can be made using four CDs or DVDs?
A) 2
B) 4
C) 6
D) 8
5. What is the least number of moves required to rearrange the bears to ensure no bear is next to a bear of the same color?
A) 2
B) 4
C) 6
D) 8
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CC-MAIN-2020-16/segments/1585370499280.44/warc/CC-MAIN-20200331003537-20200331033537-00557.warc.gz
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maths.org
|
en
| 0.919154
| 2020-03-31T02:31:03
|
https://nrich.maths.org/public/topic.php?code=-68&cl=1&cldcmpid=24
| 0.986019
|
The F-distribution is used to compare the variances of two populations, particularly in analysis of variance testing (ANOVA) and regression analysis.
**Definition 1**: The F-distribution with *n₁* and *n₂* degrees of freedom is defined as the distribution of the ratio of two independent chi-square variables, each divided by their respective degrees of freedom.
**Theorem 1**: If two independent samples of size *n₁* and *n₂* are drawn from two normal populations with the same variance, then the ratio of the sample variances follows an F-distribution with *n₁-1* and *n₂-1* degrees of freedom.
**Property 1**: A random variable *t* has a t-distribution with *k* degrees of freedom if and only if *t²* has an F-distribution with 1 and *k* degrees of freedom.
The F-distribution is used in hypothesis testing to compare variances, and is commonly used in ANOVA testing. The following Excel functions are defined for the F-distribution:
* **FDIST**(*x, df₁, df₂*) = the probability that the F-distribution with *df₁* and *df₂* degrees of freedom is greater than or equal to *x*.
* **FINV**(*α, df₁, df₂*) = the value *x* such that **FDIST**(*x, df₁, df₂*) = 1 - *α*.
In Excel 2010 and later, the following functions are also available:
* **F.DIST**(*x, df₁, df₂*, cumulative) = the cumulative F-distribution function.
* **F.INV**(*α, df₁, df₂*) = the inverse cumulative F-distribution function.
* **F.DIST.RT**(*x, df₁, df₂*) = the right-tail F-distribution function.
* **F.INV.RT**(*α, df₁, df₂*) = the inverse right-tail F-distribution function.
Note that Excel only calculates these functions for positive integer values of *df₁* and *df₂*. Non-integer values are rounded down to the nearest integer. If a more accurate value is needed for non-integer degrees of freedom, the Real Statistics noncentral F-distribution functions can be used.
The Real Statistics Resource Pack provides the following functions:
* **F_DIST**(*x*, *df₁*, *df₂*, cumulative) = a substitute for **F.DIST** that provides better estimates for non-integer degrees of freedom.
* **F_INV**(*p*, *df₁*, *df₂*) = a substitute for **F.INV** that provides better estimates for non-integer degrees of freedom.
These functions can be used to estimate the F-distribution and its inverse, and are useful for hypothesis testing and other statistical applications.
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CC-MAIN-2017-22/segments/1495463609610.87/warc/CC-MAIN-20170528082102-20170528102102-00338.warc.gz
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real-statistics.com
|
en
| 0.766464
| 2017-05-28T08:38:00
|
http://www.real-statistics.com/chi-square-and-f-distributions/f-distribution/
| 0.998397
|
# 11.2: Radical Expressions
## Learning Objectives
At the end of this lesson, students will be able to:
- Use the product and quotient properties of radicals.
- Rationalize the denominator.
- Add and subtract radical expressions.
- Multiply radical expressions.
- Solve real-world problems using square root functions.
## Vocabulary
Terms introduced in this lesson:
- radical sign
- even roots, odd roots
- simplest radical form
- rationalizing the denominator
## Key Concepts
Radicals reverse the operation of exponentiation. The index determines the kind of root. The square root is the only index which is not explicitly written but understood. Even and odd indices handle negatives differently.
## Properties of Radicals
- The product rule for radicals: The square root of the product is the product of the square roots.
- The quotient rule for radicals: The square root of the quotient is the quotient of the square roots.
- Rationalizing the denominator involves multiplying the numerator and denominator by the radical expression.
## Simplifying Radicals
To simplify radicals, ensure that:
- No fractions occur in the radicand.
- No radicals are present in the denominator of a fraction.
- The index of a radical and the exponents on any expressions in the radicand do not have common factors.
- The exponents on any expressions in the radicand are less than the index.
- The resulting expression has as few radicals as possible.
## Operations with Radicals
- When adding and subtracting radical expressions, combine only like radical terms.
- When multiplying radical expressions, multiply the numbers outside the radical sign and the numbers inside the radical sign separately, using the rule: $a\sqrt{b} \cdot c\sqrt{d}=ac\sqrt{bd}$.
## Error Troubleshooting
- Look for the highest possible perfect squares, cubes, fourth powers, etc. as indicated by the index of the radical.
- Use factor trees as guides.
- Treat constants and variables separately.
- Remind students to multiply the numbers outside the radical sign and the numbers inside the radical sign separately.
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ck12.org
|
en
| 0.752525
| 2017-04-26T22:31:31
|
http://www.ck12.org/tebook/Algebra-I-Teacher's-Edition/r1/section/11.2/
| 0.999949
|
Integer Digit Count Calculator
Calculating the number of digits in an integer is a fundamental operation in mathematics and computer science, involving determining the number of digits in a given integer. This operation is widely used in digital computing, data processing, and education to analyze or manipulate numerical data.
Historical Background
The concept of counting digits in an integer has been understood and utilized throughout the history of mathematics. Its importance increased with the advent of digital computing, where such operations are foundational in algorithms, data structures, and information processing.
Calculation Formula
The digit count of an integer \(n\) (excluding the decimal point and considering the absolute value for negative numbers) can be calculated using the formula:
\[ \text{Digit Count} = \lfloor \log_{10} n \rfloor + 1 \]
for \(n \neq 0\), and the digit count is \(1\) when \(n = 0\).
Example Calculation
To find the number of digits in the integer \(12\):
\[ \text{Digit Count} = 2 \]
And for \(22222\):
\[ \text{Digit Count} = 5 \]
Importance and Usage Scenarios
The operation of counting digits in an integer is crucial for computational tasks such as data validation, numerical analysis, and formatting numerical outputs. It is also used in educational settings to teach basic mathematical concepts and in algorithms that manipulate numbers.
Common FAQs
1. How do you handle negative integers?
The digit count is calculated based on the absolute value of the integer, so negative signs are not considered in the digit count.
2. Does this method work for decimal numbers?
This specific calculation is for integers. For decimal numbers, a different approach is needed to count the digits in the fractional part.
3. What if the integer is zero?
Zero is a special case and is considered to have 1 digit.
This calculator provides an easy and efficient way to calculate the number of digits in any given integer, facilitating users in various computational and mathematical tasks.
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CC-MAIN-2024-38/segments/1725700651722.42/warc/CC-MAIN-20240917004428-20240917034428-00184.warc.gz
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calculatorultra.com
|
en
| 0.838132
| 2024-09-17T01:46:24
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https://www.calculatorultra.com/en/tool/integer-digit-count-calculator.html
| 0.999452
|
Geometry Options
NETEXG processes polygons (boundaries), reducing all other Gerber geometry to these entities during the boolean process. For islands surrounded by metal areas, several options are available, depending on the requirements of the next software. Consider a simple ground plane with a rectangular region containing round "islands" representing via clearances. The question arises: how to describe this configuration using a collection of polygons?
All polygons must be simply connected, meaning it's possible to move from any interior point to another without crossing an edge.
Cut Line Option
A cut line is a path into and out of a polygon that enables surrounding an island. This approach is common in the IC industry and can create a polygon with multiple islands. A polygon with a cut line meets the simple connected rule and is often referred to as a reentrant polygon. However, disadvantages include:
- Complex polygons with high vertex counts, leading to slow operations
- Many CAD systems fail to process such polygons, e.g., AutoCAD cannot extrude them into 3D solids
No Cut Lines (Butting)
A second option is to break the polygon into smaller ones that don't require an island, instead butting up against each other to form the island. Disadvantages include:
- Losing the concept of one polygon per net
- Finite element programs generating an unnecessarily large number of facets where polygons abut
Leonov Polygons
A Leonov "polygon" consists of a set of polygons: a container polygon and one or more children. The rules are:
- One container polygon with a CCW boundary description
- One or more children polygons with CW boundary descriptions, which must not intersect each other or the container boundary
This description is the most compact approach and can result in the most efficient analysis if supported by finite element tools.
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CC-MAIN-2024-30/segments/1720763517541.97/warc/CC-MAIN-20240720205244-20240720235244-00492.warc.gz
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artwork.com
|
en
| 0.930653
| 2024-07-20T23:12:41
|
http://artwork.com/gerber/netex-g/geometry_options.htm
| 0.547168
|
## Types of Variable: Endogenous Variable and Exogenous Variable
Endogenous variables are used in econometrics and linear regression, similar to dependent variables. They have values determined by other variables in the system, called exogenous variables. An endogenous variable is defined as a variable whose value is determined or influenced by one or more independent variables, excluding itself.
## Endogenous Variable Example
A manufacturing plant produces a certain amount of white sugar, which is the endogenous variable, dependent on factors like weather, pests, and fuel price. The amount of sugar is entirely dependent on other factors in the system, making it purely endogenous. However, in real life, purely endogenous variables are rare, and endogenous variables are often partially determined by exogenous factors.
## Classifying Variables within a System
Identifying exogenous and endogenous variables can be challenging. Using the sugar production example, a new conveyor belt might increase sugar output. To decide if this new variable is exogenous, one must determine if the increase in output would cause the new variables to change. A variable like "weather" is definitely exogenous, as a rise in output would have no effect on the weather. However, "price" can be partially endogenous and partially exogenous, depending on the market situation.
## In Simultaneous Equations
An endogenous variable is explained by a model. In a set of simultaneous equations, the equations should explain the behavior of any endogenous variable. If the model doesn't explain the behavior of a variable, it is exogenous. For example, in a simple multiplier model with equations:
- C_t = a_1 + a_2Y_t + e_t (consumption function)
- I_t = b_1 + b_2r_t + u_t (investment function)
- Y_t = C_t + I_t + G_t (income identity function)
The variables C_t, I_t, and Y_t are endogenous, as they are explained by the model. The variables r_t (interest rate) and G_t (government spending) are exogenous, as they are not explained by the model.
## Exogenous Variables
An exogenous variable is a variable not affected by other variables in the system. Examples include weather, farmer skill, pests, and seed availability in a farming system. Exogenous variables are fixed when they enter the model, taken as a given, influence endogenous variables, and are not determined or explained by the model.
## Exogenous Variables in Experiments
In a double-blind, controlled experiment, independent variables are exogenous, as they are only affected by the researcher, who is outside the system. In other studies, independent variables may be exogenous or endogenous, which can affect the results. Controlled experiments are essential to ensure exogenous independent variables.
## Example of Simultaneous Equation
A demand and supply model can be used to illustrate exogenous and endogenous variables. For instance, the price (pt) can be endogenous if it is presented in both demand equations. To address this, techniques like Two-Stage Least Squares (TSLS) can be used, and identifying instruments for the model is crucial.
## Advice on Identifying Instruments
When working with TSLS, finding suitable instruments for the model is essential. An instrumental variable should be correlated with the endogenous variable, unaffected by the error term, and not be an intermediate variable. Tips on finding instrumental variables can be found in resources that explain instrumental variables in detail.
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CC-MAIN-2018-09/segments/1518891813622.87/warc/CC-MAIN-20180221123439-20180221143439-00499.warc.gz
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statisticshowto.com
|
en
| 0.903167
| 2018-02-21T13:36:46
|
http://www.statisticshowto.com/endogenous-variable/
| 0.626494
|
Fractals are often associated with fractal landscapes, which can generate realistic images of mountain ranges, planets, and lakes. The process of creating a fractal mountain is surprisingly simple, based on standard fractal self-symmetry with a dash of randomness. The basic idea is to start with a triangle and subdivide it into smaller triangles by dividing each edge into two sub-segments and creating a new triangle that connects the point where the sub-segments meet. By introducing randomness, this process can create compelling coastlines and mountainous landscapes.
For a two-dimensional image of a mountain, a simple randomized process, such as a randomizing L-system, can be used. The replacement rule involves subdividing a triangle into smaller triangles and randomizing the connection point of the sub-segments. Repeating this process 10 times can produce a detailed image of a mountain.
To create a three-dimensional mountain, start with a triangle and randomly move a point inside the triangle. Then, draw a line from each corner of the original triangle to that point, replacing the triangle with an irregular pyramid. Each face of the pyramid is a triangle, and repeating the process with these triangles can produce a detailed, three-dimensional mountain.
The emergence of complex structures from simple fractal rules is not surprising, as real mountains are formed by relatively simple processes such as wind, rain, and erosion. However, these processes are not simple to model in detail, and fractal landscape generation approaches the problem from a different direction.
Fractal landscapes can look realistic, but they often lack key elements such as erosion, glaciers, and volcanic action. Erosion, in particular, is difficult to capture with fractal algorithms, and post-processing is often needed to add realistic erosion effects. Benoit Mandelbrot noted that CGI landscapes often truncate at a fixed scale and are not truly fractal, and that erosion can lead to well-defined scales in real-life landscapes.
While fractals can produce impressive images, they may not fully capture the complexity of real-world landscapes. The "fractal feel" of CG landscapes can become noticeable with repeated exposure, and additional processes are needed to create more realistic environments. The relationship between simplicity and complexity in fractals is subtle, and there may be more to the universe than fractals alone.
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CC-MAIN-2023-50/segments/1700679100674.56/warc/CC-MAIN-20231207121942-20231207151942-00205.warc.gz
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goodmath.org
|
en
| 0.946515
| 2023-12-07T14:16:58
|
http://www.goodmath.org/blog/2007/09/06/fractal-mountains/
| 0.692185
|
An algorithm is an **ordered and finite set of simple operations** used to find the solution to a problem. The term originates from the classical Arabic *ḥisābu lḡubār*, meaning 'calculation using Arabic numerals'.
Algorithms enable the execution of an action or problem-solving through a series of **defined, ordered, and finite instructions**. Given an initial state and input, following the successive steps yields a final state and solution.
Algorithms are commonly used in mathematics, computer science, logic, and related disciplines, but they are also applied in everyday life to solve issues. **Examples of algorithms** include computer programs, manuals with step-by-step instructions, and recipes.
In mathematics, algorithms are used for operations like **multiplication** and **division**, as well as the **Euclid algorithm** for finding the greatest common divisor of two positive integers. Algorithms can be represented in flow charts, specifying tasks, actions, and alternatives to achieve the final goal.
A **computer algorithm** is a sequence of instructions used to solve a problem or issue in computer science or programming. All computer tasks are based on algorithms, and software or computer programs are designed using algorithms to introduce and solve tasks.
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CC-MAIN-2024-18/segments/1712296817474.31/warc/CC-MAIN-20240420025340-20240420055340-00747.warc.gz
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aviationopedia.com
|
en
| 0.919478
| 2024-04-20T05:11:34
|
https://www.aviationopedia.com/algorithm-explanations/
| 0.949167
|
### Definitions
**Voltage drop** is the amount of voltage loss in a circuit due to conductor resistance. **Conductor resistance** is determined by the conductor material, size, and ambient temperature. Voltage drop depends on the total length of conductors carrying electrical current.
In DC systems, the voltage drop length is the total round-trip distance current travels in a circuit, usually twice the length of the conductor run. In some AC systems, the distance equals the length of the conductor.
#### Reflection
Why is the conductor length different for AC and DC circuits?
The current flows constantly in DC circuits, traveling back and forth, so the distance is twice the length of a conductor. The same applies to two-wires single-phase AC systems. However, in three-wires single-phase (split-phase) and four-wires three-phase systems, the Neutral wire only returns imbalanced current, and the voltage drop calculation differs.
For split-phase systems, the voltage drop can be calculated using the two-way trip distance at 240V, similar to DC circuits.
### Voltage Drop from PV Array to Inverter
The NEC recommends a maximum voltage drop of 3%, with 2% at the DC side and 1% at the AC side. Wires should be sized to reduce resistive loss to less than 3%, which is a function of the square of the current times the resistance (I × I × R in Watts).
To choose the right wire size, use a wire-sizing table, such as the one found at Encorewire.com.
#### Example
The voltage drop formula is: Vdrop = Iop × Rc × L, where Iop is the circuit operating current, Rc is the wire's resistivity, and L is the total conductor length.
For example, given a PV array 150' away from the inverter, using #14 AWG wire with a resistivity of 3.14 Ω/kft and a current of 8.23A:
Vdrop = 8.23A × 3.14 Ω/kft × 0.3 kft = 5.168V
The voltage drop percentage is: Vdrop% = Vdrop / Vmmp = 7.75 / 357.6 = 2.16%, which exceeds the 2% limit.
Upgrading to a larger conductor size, such as #12 AWG with a resistivity of 1.98 Ω/kft:
Vdrop = 8.23A × 1.98 Ω/kft × 0.3 kft = 3.386V
While both #12 and #14 AWG conductors work for ampacity, the voltage drop calculation shows that #10 AWG is a more conservative design, although it will cost more.
Freely available online tools can be used for voltage drop calculations. If a DC option is not available, use the single-phase option and choose the correct length.
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psu.edu
|
en
| 0.856599
| 2023-09-28T20:32:07
|
https://www.e-education.psu.edu/ae868/node/967
| 0.842749
|
Considering the equation cos(z) = 4, where z is a complex variable, the goal is to solve for z. The solution involves using the definition of the cosine of a complex number and then applying the quadratic equation to solve for e^(iz).
To start, recall that cos(z) = (e^(iz) + e^(-iz))/2. Given cos(z) = 4, we can set up the equation e^(iz) + e^(-iz) = 8. Let x = e^(iz), so the equation becomes x + 1/x = 8. Multiplying both sides by x gives x^2 + 1 = 8x, or x^2 - 8x + 1 = 0. This is a quadratic equation in terms of x.
Solving the quadratic equation x^2 - 8x + 1 = 0 for x, we use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = -8, and c = 1. Substituting these values in gives x = (8 ± √((-8)^2 - 4*1*1)) / 2*1 = (8 ± √(64 - 4)) / 2 = (8 ± √60) / 2 = (8 ± 2√15) / 2 = 4 ± √15.
Therefore, e^(iz) = 4 ± √15. To solve for z, take the natural logarithm of both sides: iz = ln(4 ± √15). Then, divide both sides by i to get z = (1/i)*ln(4 ± √15) = -i*ln(4 ± √15), since 1/i = -i.
Alternatively, without converting e^(iz) to x, we can work directly with the coefficients of the equation e^(2iz) - 8e^(iz) + 1 = 0 and apply the quadratic formula to find e^(iz), yielding the same solutions.
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CC-MAIN-2017-04/segments/1484560281202.94/warc/CC-MAIN-20170116095121-00196-ip-10-171-10-70.ec2.internal.warc.gz
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openstudy.com
|
en
| 0.867011
| 2017-01-21T20:07:45
|
http://openstudy.com/updates/5061f4c6e4b0583d5cd2e44d
| 0.997196
|
An ultrafilter on a set $S$ is a collection $F$ of subsets of $S$ satisfying the axiom that for any subset $A$ of $S$, either $A$ or its complement belongs to $F$. This definition is equivalent to saying that a filter $F$ on $S$ is an ultrafilter if it is maximal among the proper filters.
In a distributive lattice, every ultrafilter is prime, and the converse holds in a Boolean algebra. An ultrafilter can also be defined as a Boolean-algebra homomorphism from a Boolean algebra $L$ to the set $\{\bot,\top\}$ of Boolean truth values.
Given an element $x$ of a set $S$, the principal ultrafilter at $x$ consists of every subset of $S$ to which $x$ belongs. An ultrafilter $F$ is fixed if the intersection of its elements is inhabited, and it is called a free ultrafilter if this intersection is empty.
The ultrafilter principle, a weak form of the axiom of choice, states that any proper filter may be extended to an ultrafilter. This principle implies that any infinite set has a free proper filter and a free ultrafilter. Free ultrafilters are important in nonstandard analysis and model theory.
There are several ways to define ultrafilters, including using the concept of codensity monads. The ultrafilter monad can be described as the codensity monad induced by the full embedding of finite sets into sets. This monad is traditionally denoted $\beta$ and is terminal among endofunctors that preserve finite coproducts.
The category of endofunctors that preserve finite coproducts is equivalent to the category of presheaves of sets on the Blass category of ultrafilters. The ultrapower functor with respect to an ultrafilter corresponds to the representable functor of that ultrafilter.
The Blass category has as objects pairs of a set $X$ and an ultrafilter $\mathcal{U}$ of $\beta X$. Morphisms are $=_{\mathcal{U}}$-equivalence classes of partial continuous maps defined on a set in $\mathcal{U}$.
Another description of ultrafilters is based on $k$-valued Post algebras. An ultrafilter on $X$ is a natural transformation $(-)^X \to (-)^1$ in the category $Prod(Fin_+, Set)$. This is equivalent to a natural transformation $(-)^X \to (-)^1$ in the functor category $Set^{Fin_+}$.
The Eilenberg-Moore category of the ultrafilter monad is the category of compact Hausdorff spaces with its obvious forgetful functor to $Set$. If $X$ is a compact Hausdorff space, the corresponding algebra structure sends an ultrafilter $F$ on $X$ to the unique point in $X$ to which $F$ converges.
Given an algebra structure $\xi\colon \beta X \to X$, a topology can be defined by declaring a set $U \subseteq X$ to be open if it is a neighborhood of each of its points. This topology is compact Hausdorff, and $\beta X$ can be equipped with a compact Hausdorff topology which is the free compact Hausdorff space generated by $X$.
The monad $\beta$ extends to the bicategory $Rel$ of sets and binary relations. The generalized multicategories defined relative to this extension can be identified with arbitrary topological spaces. Compact Hausdorff spaces are to topological spaces as monoidal categories are to multicategories.
If $X$ is a finite set, then all ultrafilters on $X$ are principal and the number of them is the cardinality of $X$. If $X$ is an infinite set of cardinality $\kappa$, then the number of ultrafilters on $X$ is $2^{2^\kappa}$.
The ultrafilter monad has been studied in various contexts, including category theory, topology, and model theory. It has connections to large cardinal hypotheses and can be used to formulate obstructions to similar codensity monads being isomorphic to the identity.
References to the ultrafilter monad can be found in the works of R. Börger, E. Manes, G. Richter, Tom Leinster, Todd Trimble, and Bill Lawvere, among others. The concept of ultrafilters has been explored in various blog posts and articles, including those by Tom Leinster and Todd Trimble.
Recent research on ultrafilters and the ultrafilter monad includes papers by Richard Garner, Jirí Adámek, and Lurdes Sousa. The ultrafilter monad remains an active area of study, with connections to topology, category theory, and model theory.
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CC-MAIN-2023-14/segments/1679296949701.0/warc/CC-MAIN-20230401032604-20230401062604-00734.warc.gz
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ncatlab.org
|
en
| 0.825016
| 2023-04-01T04:26:04
|
https://ncatlab.org/nlab/show/ultrafilter
| 0.999577
|
In this course, we will explore the world of algorithms, covering their definition, representation, and efficiency comparison. The course delves into various types of algorithms, including Brute force, Greedy, and Binary search algorithms.
#### What you will learn?
- The definition of algorithms
- Algorithm representation using flowcharts
- Comparing algorithms based on complexity
- Brute force algorithms
- Greedy algorithms
- The two pointers algorithm
- The binary search algorithm
## Course Content
### Session 1: Introduction to Brute Force
### Session 3: Greedy Algorithms
### Session 4: Two Pointers Technique
### Session 5: Binary Search Algorithm
Note: Access to this course requires a login, please enter your credentials to proceed.
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CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00053.warc.gz
|
eurekatech.org
|
en
| 0.786956
| 2024-11-03T06:57:27
|
https://eurekatech.org/courses/algorithms-101/
| 0.932046
|
Find the limits as $ x \to \infty $ and as $ x \to -\infty $ for the function $ y = x^4 - x^6 $. Use this information, together with intercepts, to give a rough sketch of the graph.
To find the limits, we first factor out $x^4$ from the function: $y = x^4(1 - x^2)$. This can be further simplified using the difference of squares formula: $y = x^4(1 - x)(1 + x)$.
As $x$ approaches infinity, we can take the limit of each term individually. The limit of $x^4$ as $x$ approaches infinity is infinity, and the limit of $1 - x^2$ is negative infinity. Since the product of a positive infinity and a negative value is negative infinity, the limit as $x$ approaches infinity is negative infinity.
As $x$ approaches negative infinity, we can again take the limit of each term individually. The limit of $x^4$ as $x$ approaches negative infinity is positive infinity, and the limit of $1 - x^2$ is negative infinity. Since the product of a positive infinity and a negative value is negative infinity, the limit as $x$ approaches negative infinity is also negative infinity.
To find the x-intercepts, we set $y$ equal to zero and solve for $x$. Factoring the function gives us $y = x^4(1 - x)(1 + x) = 0$. The solutions are $x = 0, x = 1$, and $x = -1$. The y-intercept is found by setting $x$ equal to zero, which gives us $y = 0$.
Using this information, we can sketch the graph of the function. The graph has x-intercepts at $x = 0, x = 1$, and $x = -1$, and a y-intercept at $(0, 0)$. As $x$ approaches infinity and negative infinity, the function approaches negative infinity. The graph is roughly a curve that crosses the x-axis at the intercepts and approaches negative infinity as $x$ approaches infinity and negative infinity.
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CC-MAIN-2022-05/segments/1642320303779.65/warc/CC-MAIN-20220122073422-20220122103422-00386.warc.gz
|
numerade.com
|
en
| 0.799413
| 2022-01-22T09:11:41
|
https://www.numerade.com/questions/find-the-limits-as-x-to-infty-and-as-x-to-infty-use-this-information-together-with-intercepts-to-g-2/
| 0.997923
|
The chi-square statistic measures the difference between actual and expected counts in statistical experiments, ranging from two-way tables to multinomial experiments. Actual counts come from observations, while expected counts are typically determined from probabilistic or mathematical models.
### The Formula for Chi-Square Statistic
The formula involves *n* pairs of expected and observed counts, where *e*_{k} denotes expected counts and *f*_{k} denotes observed counts. To calculate the statistic:
- Calculate the difference between corresponding actual and expected counts.
- Square these differences.
- Divide each squared difference by the corresponding expected count.
- Add the quotients to obtain the chi-square statistic.
The result is a nonnegative real number indicating the difference between actual and expected counts. A χ^{2} value of 0 indicates no difference, while a large χ^{2} value indicates disagreement between actual and expected counts.
### How to Use the Chi-Square Statistic Formula
Given data from an experiment:
- Expected: 25, Observed: 23
- Expected: 15, Observed: 20
- Expected: 4, Observed: 3
- Expected: 24, Observed: 24
- Expected: 13, Observed: 10
Compute differences by subtracting observed from expected counts:
- 25 – 23 = 2
- 15 – 20 = -5
- 4 – 3 = 1
- 24 – 24 = 0
- 13 – 10 = 3
Square and divide by expected values:
- 2^{2}/25 = 0.16
- (-5)^{2}/15 = 1.6667
- 1^{2}/4 = 0.25
- 0^{2}/24 = 0
- 3^{2}/13 = 0.6923
Add the results: 0.16 + 1.6667 + 0.25 + 0 + 0.6923 = 2.7689. Further hypothesis testing is needed to determine the significance of this χ^{2} value.
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CC-MAIN-2018-22/segments/1526794866107.79/warc/CC-MAIN-20180524073324-20180524093324-00569.warc.gz
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thoughtco.com
|
en
| 0.800987
| 2018-05-24T08:17:38
|
https://www.thoughtco.com/chi-square-statistic-formula-and-usage-3126280
| 0.995116
|
Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.
A number is both a multiple of 5 and a multiple of 6. What could this number be? To find it, we need to identify the least common multiple (LCM) of 5 and 6, which is 30.
Complete the following expressions so that each one gives a four-digit number as the product of two two-digit numbers and uses the digits 1 to 8 once and only once.
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. To double the area, we need to increase the side length by a factor of √2. Since we are adding pebbles, we can calculate the number of pebbles added each time.
The number 1,000,000 can be expressed as the product of three positive integers in several ways. We need to find all the possible combinations of three factors that multiply to 1,000,000.
Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod). This problem involves finding the difference in length between two lines of rods.
There are 1200 integers between 1 and 1200. To find the number of integers that are NOT multiples of any of the numbers 2, 3, or 5, we can use the principle of inclusion-exclusion.
This article explores divisibility tests and how they work. An article to read with pencil and paper to hand, it delves into the patterns and rules that govern divisibility.
Factor track is a game of skill where the goal is to go around the track in as few moves as possible, keeping to the rules. The game requires strategic thinking and an understanding of factors and multiples.
What is the remainder when 2^2002 is divided by 7? To find the remainder, we can look for patterns in the powers of 2 when divided by 7.
6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4, since (6!) / (2^4) = 45. We can apply this concept to find the highest power of two that divides exactly into 100!.
Can you work out what size grid you need to read our secret message? This problem involves using patterns and codes to decipher a hidden message.
What is the smallest number of answers you need to reveal in order to work out the missing headers? This problem requires strategic thinking and an understanding of how to use given information to deduce missing data.
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5, or 7? We can use the principle of inclusion-exclusion to find the number of integers that meet these conditions.
The five-digit number A679B, in base ten, is divisible by 72. To find the values of A and B, we need to consider the divisibility rules for 72.
Can you find any perfect numbers? A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding the number itself.
Find the highest power of 11 that will divide into 1000! exactly. To solve this problem, we need to consider the prime factorization of 1000! and the powers of 11 that divide into it.
Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit? This problem involves finding patterns and relationships between different variables.
Which pairs of cogs let the colored tooth touch every tooth on the other cog? Which pairs do not let this happen? Why? This problem involves understanding the relationships between different cog sizes and how they interact.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors? To solve this problem, we need to consider the prime factorization of numbers and how it relates to the number of factors.
Given the products of adjacent cells, can you complete this Sudoku? This problem requires using logic and reasoning to fill in the missing values.
The number 8888...88M9999...99 is divisible by 7, and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M? To solve this problem, we need to apply the divisibility rule for 7.
Data is sent in chunks of two different sizes - a yellow chunk has 5 characters, and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and blue chunks. We need to find the possible combinations of chunks that can fill the data slot.
Is there an efficient way to work out how many factors a large number has? To solve this problem, we need to consider the prime factorization of the number and how it relates to the number of factors.
I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5, or 6 and always have one left over. How many eggs were in the basket? This problem involves finding the least common multiple (LCM) of the given numbers.
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A? To solve this problem, we need to apply algebraic manipulations and consider the properties of digits.
Can you find what the last two digits of the number $4^{1999}$ are? To solve this problem, we need to look for patterns in the last two digits of powers of 4.
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true? We need to consider the properties of multiples of 6 and how they relate to the number of factors.
A number N is divisible by 10, 90, 98, and 882 but is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N? To solve this problem, we need to apply the divisibility rules and consider the factors of the given numbers.
Find the number which has 8 divisors, such that the product of the divisors is 331776. To solve this problem, we need to consider the properties of numbers with a given number of divisors and how they relate to the product of the divisors.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares? To solve this problem, we need to consider the properties of factorials and perfect squares.
The clues for this Sudoku are the product of the numbers in adjacent squares. This problem requires using logic and reasoning to fill in the missing values.
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid. This problem involves using patterns and logic to deduce the missing values.
Play this game and see if you can figure out the computer's chosen number. This problem involves using strategic thinking and pattern recognition to deduce the computer's number.
Can you find a way to identify times tables after they have been shifted up or down? To solve this problem, we need to consider the properties of times tables and how they relate to the shifted tables.
Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma, and Emma passed a fifth of her counters to Ben. After this, they all had the same number of counters. We need to find the initial number of counters each person had.
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all? To solve this problem, we need to consider the properties of cuboids and how they relate to the surface area.
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it? This problem involves using mathematical induction and pattern recognition to prove the statement.
Using your knowledge of the properties of numbers, can you fill all the squares on the board? This problem requires using logic and reasoning to fill in the missing values.
Each letter represents a different positive digit AHHAAH / JOKE = HA. What are the values of each of the letters? To solve this problem, we need to apply algebraic manipulations and consider the properties of digits.
Choose any 3 digits and make a 6-digit number by repeating the 3 digits in the same order (e.g., 594594). Explain why whatever digits you choose, the number will always be divisible by 7, 11, and 13. This problem involves using pattern recognition and properties of divisibility to prove the statement.
Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I swim each time. Can you explain the strategy for winning this game with any target?
Got It game for an adult and child. How can you play so that you know you will always win? This problem involves using strategic thinking and pattern recognition to develop a winning strategy.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now? This problem involves using pattern recognition and properties of sequences to determine the possible numbers.
Factors and Multiples game for an adult and child. How can you make sure you win this game? This problem involves using strategic thinking and properties of factors and multiples to develop a winning strategy.
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was? To solve this problem, we need to apply algebraic manipulations and consider the properties of numbers.
Can you find any two-digit numbers that satisfy all of these statements? This problem involves using logic and reasoning to find the numbers that meet the given conditions.
How many noughts are at the end of these giant numbers? To solve this problem, we need to consider the properties of factors and multiples, particularly the number of trailing zeros in a given number.
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A?
A) 3
B) 5
C) 7
D) 9
The number 8888...88M9999...99 is divisible by 7, and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?
A) 1
B) 3
C) 5
D) 7
A number N is divisible by 10, 90, 98, and 882 but is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?
A) 10
B) 30
C) 90
D) 180
Find the number which has 8 divisors, such that the product of the divisors is 331776.
A) 12
B) 24
C) 36
D) 48
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
A) 1
B) 2
C) 3
D) 4
The clues for this Sudoku are the product of the numbers in adjacent squares.
What is the value of the missing digit?
A) 3
B) 5
C) 7
D) 9
Can you find what the last two digits of the number $4^{1999}$ are?
A) 04
B) 16
C) 24
D) 36
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?
A) Yes
B) No
C) Sometimes
D) Never
A number N is divisible by 10, 90, 98, and 882 but is NOT divisible by 50 or 270 or 686 or 1764. What is N?
A) 10
B) 30
C) 90
D) 180
Find the highest power of 11 that will divide into 1000! exactly.
A) 11^1
B) 11^2
C) 11^3
D) 11^4
Can you find any perfect numbers?
A) Yes
B) No
C) Sometimes
D) Never
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
A) 12
B) 24
C) 36
D) 48
Given the products of adjacent cells, can you complete this Sudoku?
What is the value of the missing digit?
A) 3
B) 5
C) 7
D) 9
The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
What is the value of the missing digit?
A) 3
B) 5
C) 7
D) 9
Play this game and see if you can figure out the computer's chosen number.
What is the computer's chosen number?
A) 10
B) 20
C) 30
D) 40
Can you find a way to identify times tables after they have been shifted up or down?
A) Yes
B) No
C) Sometimes
D) Never
Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma, and Emma passed a fifth of her counters to Ben. After this, they all had the same number of counters.
How many counters did Ben have initially?
A) 10
B) 20
C) 30
D) 40
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units.
What are the dimensions of the cuboid?
A) 5x5x4
B) 4x5x5
C) 5x4x5
D) 4x4x6
List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3.
Can you explain why and prove it?
A) Yes
B) No
C) Sometimes
D) Never
Using your knowledge of the properties of numbers, can you fill all the squares on the board?
What is the value of the missing digit?
A) 3
B) 5
C) 7
D) 9
Each letter represents a different positive digit AHHAAH / JOKE = HA.
What are the values of each of the letters?
A) A=1, H=2, J=3, O=4, K=5, E=6
B) A=2, H=3, J=1, O=4, K=5, E=6
C) A=3, H=1, J=2, O=4, K=5, E=6
D) A=4, H=2, J=3, O=1, K=5, E=6
Choose any 3 digits and make a 6-digit number by repeating the 3 digits in the same order (e.g., 594594).
Explain why whatever digits you choose, the number will always be divisible by 7, 11, and 13.
A) Yes
B) No
C) Sometimes
D) Never
Twice a week I go swimming and swim the same number of lengths of the pool each time.
Can you explain the strategy for winning this game with any target?
A) Yes
B) No
C) Sometimes
D) Never
Got It game for an adult and child.
How can you play so that you know you will always win?
A) Yes
B) No
C) Sometimes
D) Never
Imagine we have four bags containing numbers from a sequence.
What numbers can we make now?
A) 1, 2, 3, 4
B) 2, 4, 6, 8
C) 1, 3, 5, 7
D) 3, 6, 9, 12
Factors and Multiples game for an adult and child.
How can you make sure you win this game?
A) Yes
B) No
C) Sometimes
D) Never
Gabriel multiplied together some numbers and then erased them.
Can you figure out where each number was?
A) Yes
B) No
C) Sometimes
D) Never
Can you find any two-digit numbers that satisfy all of these statements?
A) 12
B) 24
C) 36
D) 48
How many noughts are at the end of these giant numbers?
A) 1
B) 2
C) 3
D) 4
|
CC-MAIN-2019-09/segments/1550247490806.45/warc/CC-MAIN-20190219162843-20190219184843-00020.warc.gz
|
maths.org
|
en
| 0.897033
| 2019-02-19T16:34:09
|
https://nrich.maths.org/public/leg.php?code=12&cl=3&cldcmpid=5339
| 0.998733
|
# Mean-Variance Ceiling
While analyzing count data from a small RNA-Seq experiment in *Arabidopsis thaliana*, a notable pattern emerged in the mean-variance relationship for fragment counts. Despite the small dataset, with only 3 replicates per condition and each sample from a different batch, a clear straight line was visible in the mean-variance plot, representing a ceiling that no points crossed.
The sample variance of \(n\) numbers \(a_1,\ldots,a_n\) is given by \(\sigma^2=\frac{n}{n-1}\left(\frac1n\sum_{i=1}^n a_i^2-\mu^2\right)\), where \(\mu\) is the sample mean. This can be further simplified to \(\frac{\sigma^2}{\mu^2}=\frac{\sum a_i^2}{(n-1)\mu^2}-\frac{n}{n-1}\).
For non-negative numbers, the relationship \(n^2\mu^2=(\sum a_i)^2\geq \sum a_i^2\) holds, leading to \(\frac{\sigma^2}{\mu^2}\leq\frac{n^2}{n-1}-\frac{n}{n-1}=n\). On a log-log plot, this translates to all points \((\mu,\sigma^2)\) lying on or below the line \(y=2x+\log n\).
The points exactly on this line correspond to samples where all \(a_i\) but one are zero, indicating gene-condition combinations where a gene's transcripts were registered in only one replicate for that condition. This phenomenon explains the observed ceiling in the mean-variance plot.
|
CC-MAIN-2023-23/segments/1685224652184.68/warc/CC-MAIN-20230605221713-20230606011713-00765.warc.gz
|
ro-che.info
|
en
| 0.919464
| 2023-06-05T23:50:05
|
https://ro-che.info/articles/2016-10-20-mean-variance-ceiling
| 0.991314
|
## What is a Proportion?
A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls, you could write the ratio as 1:3 (for every one boy there are 3 girls) or 1/4 are boys and 3/4 are girls.
## How to Write a Ratio
To write a ratio, determine whether it is part to part or part to whole, calculate the parts and the whole if needed, plug values into the ratio, and simplify the ratio if needed. Integer-to-integer ratios are preferred.
## How to Identify a Proportion
Ratios are proportional if they represent the same relationship. One way to see if two ratios are proportional is to write them as fractions and then reduce them. If the reduced fractions are the same, your ratios are proportional.
## Proportion Formula
The proportion formula is used to depict if two ratios or fractions are equal. We can find the missing value by dividing the given values. The proportion formula can be given as a:b::c:d = a/b = c/d, where a and d are the extreme terms and b and c are the mean terms.
## Examples of Proportions
- 16:24 = 20:30 is true.
- 4/5 and 16:20 are in proportion.
- 16, 30, 24, 45 are in proportion.
- 12, 15, 4, 5 are in proportion.
- 2:3 = 4:6, so 2, 3, 4, and 6 are in proportion.
- 5/6, 15/18 are in proportion.
- 15, 45, 40, 120 are in proportion.
## What are the 4 Terms of a Proportion?
The four numbers a, b, c, and d are known as the terms of a proportion. The first (a) and the last term (d) are referred to as extreme terms, while the second and third terms in a proportional are called mean terms.
## Testing Proportions
To test if two ratios are proportional, simplify each ratio to its simplest form. If the simplified fractions are the same, the proportion is true; if the fractions are different, the proportion is false.
## Proportional Relationship
A proportional relationship between two quantities is a collection of equivalent ratios, related to each other by a constant of proportionality. Proportional relationships can be represented in different, related ways, including a table, equation, graph, and written description.
## Identifying Proportional Relationships
To identify if two ratios are in proportion, check if their simplest forms are equal. For example, 12:15 and 18:20 are not in proportion because their simplest forms (4:5 and 9:10) are not equal.
## Proportion in Real Life
Proportion refers to the dimensions of a composition and relationships between height, width, and depth. How proportion is used will affect how realistic or stylized something seems. Proportion also describes how the sizes of different parts of a piece of art or design relate to each other.
## Multiple Choice Questions
1. Are 16:24 and 20:30 in proportion?
- A) Yes
- B) No
Answer: A) Yes
2. Is 4/5 proportional to 16:20?
- A) Yes
- B) No
Answer: A) Yes
3. Are 12, 15, 4, 5 in proportion?
- A) Yes
- B) No
Answer: A) Yes
4. What is the proportion formula?
- A) a:b::c:d = a/b = c/d
- B) a:b::c:d = a/b ≠ c/d
Answer: A) a:b::c:d = a/b = c/d
5. Are 5/6, 15/18 in proportion?
- A) Yes
- B) No
Answer: A) Yes
|
CC-MAIN-2023-14/segments/1679296950363.89/warc/CC-MAIN-20230401221921-20230402011921-00544.warc.gz
|
tissfla.com
|
en
| 0.913999
| 2023-04-01T22:25:33
|
https://tissfla.com/articles/are-16-24-20-30-are-in-proportion
| 0.998732
|
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