what is the maximum value of the function?

since we have found an element (48) in S that is not in T, we can conclude that S is not equal to T.

To prove that S is not equal to T, we need to show that there I an element in either S or T that is not in the other set.

Let's first look at the elements in S. We know that S is the set of all integers that can be expressed as 24 times some other integer. So, for example, 24, 48, 72, -24, -48, -72, etc. are all in S.

Now, let's look at the elements in T. We know that T is the set of all integers that can be expressed as 4 times some integer and 6 times some integer. We can find some examples of numbers in T by finding the multiples of the LCM of 4 and 6, which is 12. So, for example, 12, 24, 36, -12, -24, -36, etc. are all in T.

Now, let's consider the number 48. We know that 48 is in S, since it can be expressed as 24 times 2. However, 48 is not in T, since it cannot be expressed as 4 times some integer and 6 times some integer. This is because the only common multiple of 4 and 6 is 12, and 48 is not a multiple of 12.

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To find the probability of selecting a person with blood type AB from a random distribution of 100 Americans, some steps need to be followed.

Steps are:
Step 1: Identify the total number of people (100 Americans in this case) and the number of people with blood type AB from the table (AB+ and AB-).

Step 2: Add the number of people with AB+ and AB- blood types:
AB+ (2) + AB- (4) = 6

Step 3: Calculate the probability by dividing the number of people with blood type AB (6) by the total number of people (100):
Probability = (Number of AB blood types) / (Total number of people)
Probability = 6 / 100

Step 4: Simplify the fraction to get the final probability:
Probability = 0.06

So, the probability of selecting a person with blood type AB from a random distribution of 100 Americans is 0.06 or 6%.

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Our estimate is that around 40,514 peas can fit inside a 1 gallon jar under these assumptions.

The number of peas that fit inside a 1 gallon jar can vary depending on a few factors, such as the size of the peas, the packing density, and the shape of the jar. However, we can make a rough estimate based on some assumptions and calculations.

Assuming that the peas are spherical and have an average diameter of 0.5 cm, we can calculate the volume of each pea using the formula for the volume of a sphere:

[tex]V = (4/3)πr^3[/tex]

where r is the radius of the sphere, which is half the diameter. Thus, for a pea with a diameter of 0.5 cm, the radius is 0.25 cm, and the volume is:

V = (4/3)π(0.25 cm)^3 ≈ 0.0654 [tex]cm^3[/tex]

Next, we need to estimate the volume of the 1 gallon jar. One gallon is equal to 3.78541 liters, or 3785.41 cubic centimeters (cc). However, the jar may not be filled to its full volume due to its shape and the presence of the peas, so we need to make an assumption about the packing density. Let's assume that the peas occupy 70% of the volume of the jar, leaving 30% as empty space. This gives us an estimated volume of:

V_jar = 0.7(3785.41 cc) ≈ 2650.79 cc

To find the number of peas that fit inside the jar, we can divide the estimated volume of the jar by the volume of each pea:

N = V_jar / V ≈ 40,514

Therefore, our estimate is that around 40,514 peas can fit inside a 1 gallon jar under these assumptions. It's important to note that this is only an approximation, and the actual number may vary depending on the factors mentioned earlier.

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The value of x that satisfies the given conditions is approximately 7.

Given that KLMN is similar to PQRS. This means that the corresponding sides of these two angles are proportional. We can use this property to set up a proportion between the sides of the two angles.

Let LM and RQ be corresponding sides in the two triangles, and let KN and SP be the other corresponding sides. Then we have:

LM/RQ = KN/SP

Substituting the given values, we get:

(4x + 4)/48 = (7x - 9)/60

To solve for x, we cross-multiply, which gives us:

(4x + 4) * 60 = (7x - 9) * 48

Expanding both sides, we get:

336x - 432 = 240x + 240

Simplifying and solving for x, we get:

96x = 672

x = 7

Therefore, the value of x that satisfies the given conditions is approximately 7.

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The equation of the parabola is given as follows:

y = -16/33124(x - 182)² + 26.

The distance from the center at which the height is 16 feet is given as follows:

38.12 ft and 325.88 ft.

The equation of a parabola of vertex (h,k) is given by the equation presented as follows:

y = a(x - h)² + k.

In which a is the leading coefficient.

It has a span of 364 feet, hence the x-coordinate of the vertex is given as follows:

x = 364/2

x = 182.

It has a maximum height of 26 feet, hence the y-coordinate of the vertex is obtained as follows:

y = 26.

Considering that h = 182 and k = 26, the equation is:

y = a(x - 182)² + 26.

When x = 0, y = 0, hence the leading coefficient a is obtained as follows:

33124a + 26 = 0

a = -26/33124

Hence:

y = -16/33124(x - 182)² + 26.

For a height of 16 feet, we have that

y = 16

16/33124(x - 182)² = 10

(x - 182)² = 33124 x 10/16

(x - 182)² = 20702.5.

Hence the heights are:

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The behavior of the integral of [tex]2e^(-x)[/tex] dx, given by the function [tex]-2e^(-x) + C[/tex], shows an exponential decay.

To algebraically determine the behavior of the integral of [tex]2e^(-x) dx[/tex], we need to perform the integration and observe the resulting function.

Step 1: Integrate the function with respect to x:
We want to find the integral ∫[tex]2e^(-x) dx[/tex]. To do this, we apply the integration rule ∫[tex]e^(ax) dx = (1/a)e^(ax) + C[/tex], where a is a constant and C is the integration constant.

In our case, a = -1. So, the integral becomes:
∫[tex]2e^(-x) dx = (1/-1) * 2e^(-x) + C = -2e^(-x) + C[/tex]

Step 2: Analyze the behavior of the function:
Now that we have the integral, we can observe its behavior. The resulting function is [tex]-2e^(-x) + C[/tex], which is an exponential decay function with a negative coefficient. As x approaches positive infinity, [tex]e^(-x)[/tex] approaches 0, making the function approach the constant value C. Similarly, as x approaches negative infinity, [tex]e^(-x)[/tex] approaches infinity, making the function approach negative infinity.

In summary, the behavior of the integral of [tex]2e^(-x)[/tex] dx, given by the function [tex]-2e^(-x) + C[/tex], shows an exponential decay. As x increases, the function approaches a constant value, while as x decreases, the function approaches negative infinity. This behavior is due to the negative coefficient and the exponential term [tex]e^(-x)[/tex] in the function.

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The behavior of the integral of [tex]2e^(-x)[/tex] dx, given by the function [tex]-2e^(-x) + C[/tex], shows an exponential decay.

To algebraically determine the behavior of the integral of [tex]2e^(-x) dx[/tex], we need to perform the integration and observe the resulting function.

Step 1: Integrate the function with respect to x:
We want to find the integral ∫[tex]2e^(-x) dx[/tex]. To do this, we apply the integration rule ∫[tex]e^(ax) dx = (1/a)e^(ax) + C[/tex], where a is a constant and C is the integration constant.

In our case, a = -1. So, the integral becomes:
∫[tex]2e^(-x) dx = (1/-1) * 2e^(-x) + C = -2e^(-x) + C[/tex]

Step 2: Analyze the behavior of the function:
Now that we have the integral, we can observe its behavior. The resulting function is [tex]-2e^(-x) + C[/tex], which is an exponential decay function with a negative coefficient. As x approaches positive infinity, [tex]e^(-x)[/tex] approaches 0, making the function approach the constant value C. Similarly, as x approaches negative infinity, [tex]e^(-x)[/tex] approaches infinity, making the function approach negative infinity.

In summary, the behavior of the integral of [tex]2e^(-x)[/tex] dx, given by the function [tex]-2e^(-x) + C[/tex], shows an exponential decay. As x increases, the function approaches a constant value, while as x decreases, the function approaches negative infinity. This behavior is due to the negative coefficient and the exponential term [tex]e^(-x)[/tex] in the function.

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Thus, the simplification of the given polynomial is given as;

-68u²v²  - 2u⁸v⁴.

The tight definition makes polynomials simple to work with.

For instance, we are aware of:

One-variable polynomials are very simple to graph due to their smooth, continuous lines.

The biggest exponent of a polynomial with a single variable is the polynomial's degree.

For the given polynomial:

-71uv²u + (3vu²v - 5u⁶u²v⁴) + 3u³v²v²u⁵

Open the brackets:

-71uv²u + 3vu²v - 5u⁶u²v⁴ + 3u³v²v²u⁵

The powers with the same base get added with sign:

-71u¹⁺¹ v² + 3v¹⁺¹ u² - 5u⁶⁺² v⁴ + 3u³⁺⁵ v²⁺²

-71u² v² + 3v² u² - 5u⁸v⁴ + 3u⁸ v⁴

The coefficients with the same variable gets added with sign:

(-71u² v² + 3v² u²) + (- 5u⁸v⁴ + 3u⁸ v⁴ )

(-68u²v² ) + (- 2u⁸v⁴)

-68u²v²  - 2u⁸v⁴

Thus, the simplification of the given polynomial is given as;

-68u²v²  - 2u⁸v⁴.

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Complete question:

Simplify the polynomial:

-71uv²u + (3vu²v - 5u⁶u²v⁴) + 3u³v²v²u⁵

The differential dy of the function y = 2x^4 - 54 - 4x is dy = (8x^3 - 4)dx.

To find the differential dy of the function y = 2x^4 - 54 - 4x, we first need to differentiate y with respect to x.

Step 1: Identify the terms in the function. The terms are 2x^4, -54, and -4x.

Step 2: Differentiate each term with respect to x.
- For 2x^4, using the power rule (d/dx (x^n) = n*x^(n-1)), we get (4)(2x^3) = 8x^3.
- For -54, since it's a constant, its derivative is 0.
- For -4x, using the power rule, we get (-1)(-4x^0) = -4.

Step 3: Combine the derivatives to get the derivative of the entire function.
dy/dx = 8x^3 - 4.

Step 4: The differential dy is the derivative multiplied by dx.
dy = (8x^3 - 4)dx.

So, the differential dy of the function y = 2x^4 - 54 - 4x is dy = (8x^3 - 4)dx.

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The temperature ratio T2/T1 = I^(γ-1), and for small disturbances, the change in entropy ΔS ≈ n * R * 2 * (γ - 1) * ε.

The temperature ratio T2/T1 in terms of the pressure ratio P2/P1 can be found using the adiabatic process equation for an ideal gas. For an adiabatic process, the equation is:

P1 * V1^γ = P2 * V2^γ

Where γ is the heat capacity ratio (Cp/Cv). Since P2/P1 = I, we can rewrite the equation as:

V1^γ = V2^γ * I

From the ideal gas law, we know that P1 * V1 / T1 = P2 * V2 / T2, so:

V1 / T1 = V2 / T2 * I

Now, we can substitute V1^γ from the first equation into the second equation:

T2 / T1 = I^(γ-1)

For the change in entropy, we can use the formula:

ΔS = n * Cv * ln(T2 / T1) + n * R * ln(V2 / V1)

Substituting the temperature ratio and the volume ratio, we get:

ΔS = n * R * [(γ - 1) * ln(I) + ln(I^(γ - 1))]

For small disturbances, where I = 1 + ε and ε << 1, we can use the approximation ln(1 + ε) ≈ ε:

ΔS ≈ n * R * (γ - 1) * ε + n * R * (γ - 1) * ε

ΔS ≈ n * R * 2 * (γ - 1) * ε

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Answer:

B and E

Step-by-step explanation:

They both have the same number of untis as Z

If the coordinate of an object is given as a function of time by 0 = 7t-3t2, where is in radians and t is in seconds, The angular velocity at t = 3 s is -11 rad/s. The answer is (a) -11 rad/s.

The angular velocity is the derivative of the position function with respect to time. Therefore, we need to find the derivative of the given function:

θ = 7t - 3t^2

ω = dθ/dt = 7 - 6t

Now we can find the angular velocity at t = 3 s by plugging in t = 3 into the equation for ω:

ω = 7 - 6(3) = -11

Therefore, The answer is (a) -11 rad/s.

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If the coordinate of an object is given as a function of time by 0 = 7t-3t2, where is in radians and t is in seconds, The angular velocity at t = 3 s is -11 rad/s. The answer is (a) -11 rad/s.

The angular velocity is the derivative of the position function with respect to time. Therefore, we need to find the derivative of the given function:

θ = 7t - 3t^2

ω = dθ/dt = 7 - 6t

Now we can find the angular velocity at t = 3 s by plugging in t = 3 into the equation for ω:

ω = 7 - 6(3) = -11

Therefore, The answer is (a) -11 rad/s.

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The set {[tex]v_{1 }, v_{2}, v_{3}[/tex]}  is the basis for the subspace of R4 because C1=C2=C3=0.

What is a subspace?

It is a part of linear algebra. The members of the subspace are all vectors and also they all have same dimensions. It is also called as vector subspace. A vector space that is totally contained within another vector space is known as a subspace. Both are required to completely define one because a subspace is defined relative to its contained space; for instance, R2 is a subspace of R3, but also of R4, C2, etc.

The given set in the question is:

{(1,-2,3,4),(-1,3,0,-2),(2,-3,9,10)}

As the set {V1, V2, V3} spam a subset of R4;

then,

C1V1 + C2V2 + C3V3= 0

C1(1,-2,3,4) + C2(-1,3,0,-2) + C3(2,-3,9,10) =0

On solving we will get following equation from above equation:

C1 + 2C2 + C3 =0

C1-C3=0

-5C1 + 2C2=0

-6C1 - 2C2 + 8C3 =0

From the above equation we can easily conclude that;

C1=C2=C3=0

So, {V1,V2,V3}  are linearly independent.

Thus set is the basis for subspace of R4.

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The radius of convergence R, we use the Ratio Test: R = lim (n→∞) |(aₙ₊₁ / aₙ)|.

The Taylor series for f centered at 8 is given by the formula:

Σ[(-1)ⁿ * (n! * (x-8)ⁿ) / (4ⁿ * (n+2)ⁿ)], where n ranges from 0 to infinity.

The radius of convergence R is 1/4.

To find the Taylor series, we use the general formula for Taylor series expansion:

Σ[(fⁿ(8) * (x-8)ⁿ) / n!], where n ranges from 0 to infinity.

Given that fⁿ(8) = (-1)ⁿ * n! / 4ⁿ * (n+2)ⁿ, we substitute this into the Taylor series formula:

Σ[((-1)ⁿ * n! / 4ⁿ * (n+2)ⁿ) * (x-8)ⁿ / n!] = Σ[(-1)ⁿ * (x-8)ⁿ / (4ⁿ * (n+2)ⁿ)].

To find the radius of convergence R, we use the Ratio Test:

R = lim (n→∞) |(aₙ₊₁ / aₙ)|.

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Answer:

Step-by-step explanation:

24/5-12/5=12/5

To simplify 2 4/5 - 1 2/5, we first need to convert the mixed numbers into improper fractions.

Step 1: Convert the mixed numbers to improper fractions:

2 4/5 = (2 x 5 + 4)/5 = 14/5

1 2/5 = (1 x 5 + 2)/5 = 7/5

Step 2: Subtract the two improper fractions:

14/5 - 7/5 = (14 - 7)/5 = 7/5

Step 3: Convert the resulting fraction back into a mixed number, if necessary:

7/5 can be written as 1 2/5

Therefore, 2 4/5 - 1 2/5 = 1 2/5

Answer:

multiply the bottom by 2

Step-by-step explanation:

Answer:

Step-by-step explanation:

The equation of the line that passes through the points (3, 6) and (-1, -4) can be found using the point-slope formula.

First, find the slope of the line using the formula:

slope = (y2 - y1)/(x2 - x1)

where (x1, y1) = (3, 6) and (x2, y2) = (-1, -4).

slope = (-4 - 6)/(-1 - 3) = -10/-4 = 5/2

Now that we have the slope, we can use it in the point-slope formula:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is either one of the given points. Let's use (3, 6):

y - 6 = (5/2)(x - 3)

Simplifying this equation, we get:

y - 6 = (5/2)x - 15/2

y = (5/2)x - 3/2

Therefore, the equation of the line that passes through the points (3, 6) and (-1, -4) is y = (5/2)x - 3/2.

Answer:

5/2

Step-by-step explanation:

Slope = change in y coordinate/change in x coordinate.

In this example, Slope = [tex]\frac{-4 - 6}{-1 - 3} = \frac{-10}{-4} = \frac{10}{4} =\frac{5}{2}[/tex]

Your slope is 5/2.

The area of the regular pentagon is 558 ft².

The area of the regular hexagon is 374.12 in².

The area of the regular polygon is calculated as follows;

A = ¹/₂ Pa

where;

The perimeter of the regular polygon is calculated as follows;

P = 18 ft x 5

P = 90 ft

The area of the regular pentagon is calculated as;

A =  ¹/₂ Pa

A = ¹/₂ x 90 ft  x 12.4 ft

A = 558 ft²

The area of the regular hexagon is calculated as;

A = a² x 3√3 / 2

where;

A = 12² in x 3√3 / 2

A = 374.12 in²

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The difference between the minimum and the maximum value of the expression 14a^2b^2 is 224.

The maximum of the quantity 14a^2b^2 occurs from the given equation, we know that a = 4/b. Substituting this into the expression for 14a^2b^2, we get:

14(4/b)^2b^2 = 14(16/b^2)*b^2

                     =224

So the maximum value of 14a^2b^2 is 224, here a, b is non-negative integers, to get the minimum value of the expression is one of the integer must be zero if the one of the integers is zero then the minimum value of the expression is becomes 0.

Explanation; -

STEP 1:- To get the maximum value of the function use the given conditions a b=4 and substitute in the given expression 14a^2b^2.

STEP2:-  After substituting the value evaluate the expression and get the maximum value of the expression.

STEP3:-  To get the minimum value of the expression minimize the value of the a and b by the observation it is clear that the minimum value of the expression is zero.

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The Bohr model is not an accurate representation of the hydrogen atom, as the actual probability density function for the 2p state has a maximum at a larger distance from the nucleus than predicted by the Bohr model.

To find the value of r at which P(r) is a maximum, we need to differentiate the expression for P(r) with respect to r and set it equal to zero:

d[P(r)]/dr = 2R2p² r - 4R2p² r²/a = 0

Simplifying and solving for r, we get:

r = 2a/3

Substituting this value of r back into the expression for P(r), we get:

P(r) = (R2p)² (2a/3)²

P(r) = (1/24a⁵) e^(-2/3) (2a/3)⁴

P(r) = (16/81πa³) e^(-2/3)

To compare this result to the radius of the n=2 state in the Bohr model, we can use the expression for the Bohr radius:

a0 = 4πε0 ħ²/m_e e²

a0 = 0.529 Å

The maximum value of P(r) for the 2p state occurs at a distance of 2a/3 from the nucleus, which is approximately 0.88 Å. This is larger than the Bohr radius for the n=2 state, which is 0.529 Å.

Therefore, we can see that the Bohr model is not an accurate representation of the hydrogen atom, as the actual probability density function for the 2p state has a maximum at a larger distance from the nucleus than predicted by the Bohr model.

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The equation of the curve is y = x² - 4x + 3

Since we are given the slope of the curve at each point, we can use integration to find the equation of the curve. Let's denote the equation of the curve as y = f(x).

The slope of the curve is given by dy/dx = 2x - 4. We can integrate this expression with respect to x to obtain an expression for f(x):

∫dy = ∫(2x - 4)dx

y = x² - 4x + C

where C is the constant of integration.

To determine the value of C, we use the fact that the curve passes through the point (4,7):

7 = 4² - 4(4) + C

C = 7 + 4(4) - 16 = 3

Thus, the equation of the curve is y = x²- 4x + 3.

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E(y) = 1.

We know that:

z = log(y)

Taking the exponential of both sides, we get:

e^z = y

Now, we want to find E(y). We can use the definition of expected value:

E(y) = ∫y*f(y)dy

where f(y) is the probability density function of y. To find f(y), we use the change of variables formula:

f(y) = f(z) * |dz/dy|

where f(z) is the probability density function of z, which is the standard normal distribution, and |dz/dy| is the absolute value of the derivative of z with respect to y:

dz/dy = 1/y

|dz/dy| = 1/y

Substituting in the expression for f(y), we get:

f(y) = f(z) * (1/y)

The density function of the standard normal distribution is:

f(z) = (1/√(2π)) * e^(-z^2/2)

Substituting this expression and the expression for y in terms of z, we get:

f(y) = (1/√(2π)) * e^(-(log(y))^2/2) * (1/y)

We can now plug this expression into the formula for E(y):

E(y) = ∫y*f(y)dy

= ∫e^z * (1/√(2π)) * e^(-(log(y))^2/2) * (1/y) dy

= ∫e^(z - (log(y))^2/2) * (1/√(2π)) dz [using the fact that dy/y = dz]

= ∫e^(-(log(y))^2/2) * (1/√(2π)) dz [since e^z is integrated over the entire range of z]

= (1/√(2π)) * ∫e^(-z^2/2) dz [using the substitution z = log(y)]

= (1/√(2π)) * √(2π) [using the fact that ∫e^(-z^2/2) dz is the integral of the standard normal density function over its entire domain, which is equal to 1]

= 1

Therefore, E(y) = 1.

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In order for the characteristics of a sample to be generalized to the entire population, the sample should be option (c)  representative of the population

For a sample to be able to generalize to the entire population, it must be selected in such a way that it accurately reflects the characteristics of the population from which it was drawn. This means that the sample should be representative of the population in terms of the relevant characteristics that are being studied.

If the sample is not representative of the population, then any conclusions drawn from the sample may not be applicable to the larger population, which can lead to inaccurate or misleading results.

Therefore, it is important to use proper sampling methods to ensure that the sample is representative of the population. This can be done through techniques such as random sampling or stratified sampling, which aim to select a sample that accurately reflects the population characteristics of interest.

Therefore, the correct option is (c) representative of the population.

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The limit of the function [(4x² + 10y² + 4) / (4x² - 10y² + 6)] as (x, y) approaches (0, 0) is 2/3.

In mathematics, a limit is a value that a function approaches as the input approaches some value.

To evaluate the limit of the given function at the point (0, 0), we have the following expression:
Limit as (x, y) approaches (0, 0) of [(4x² + 10y² + 4) / (4x² - 10y² + 6)].

Substitute x = 0 and y = 0 into the given expression:
[(4(0)² + 10(0)² + 4) / (4(0)² - 10(0)² + 6)] = [4 / 6].

Simplify the expression:
4 / 6 = 2 / 3.

So, the limit of the given function as (x, y) approaches (0, 0) is 2/3. The limit exists, and its value is 2/3.

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There were 200 participants at the telematch.

The second degree is represented mathematically by a quadratic equation, where the highest power of the variable is 2.

It is expressed as ax² + bx + c = 0, where x is the variable and a, b, and c are the coefficients.

Let the total number of participants be P. Then, the number of children is (P-125), and the number of girls is (P-125) × (1-B/(P-125)), where B is the number of boys, put all values:

(P-125) × (1-B/(P-125)) = (P-B-125)/2

Simplifying the above equation, we get:

B² - 250B + (P-125)² = 0

We know the quadratic formula;

B = (250 ± √(250² - 4×(P-125)²))/2

Since B must be an integer, only the positive root is possible, and it must be a whole number.
Therefore, we can solve for P by trying out integer values for B until we find one that gives a whole number for P. Trying out values, we find that B = 100 gives P = 200, which is a whole number.
Therefore, there were 200 participants at the Telematch.

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A key difference between MANOVA and ANOVA is that MANOVA has the ability to handle multiple dependent variables.

The key difference between MANOVA and ANOVA is that MANOVA has the ability to handle multiple dependent variables. ANOVA, on the other hand, only deals with a single dependent variable. MANOVA can also handle multiple independent variables, multiple error terms, and multiple null hypotheses. However, it is important to note that MANOVA is more complex than ANOVA and requires more data to perform accurate analyses.

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Answer:It is given b the x2 of the anwserr fot the firghjt angle

Step-by-step explanation:

dz/dt for the given functions is [tex]16a^3t^3 + 10a^3t^4[/tex].

To find dz/dt for z(x, y) = [tex]xy^2 + x^2y[/tex], x(t) = at^2, and y(t) = 2at, we'll use the chain rule.

Here's a step-by-step explanation:

Step 1: Find the partial derivatives of z with respect to x and y. [tex]∂z/∂x = y^2 + 2xy ∂z/∂y = 2xy + x^2[/tex]

Step 2: Find the derivatives of x(t) and y(t) with respect to t. dx/dt = 2at dy/dt = 2a

Step 3: Apply the chain rule to find dz/dt. dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

Step 4: Substitute the expressions from steps 1 and 2 into the chain rule equation. dz/dt = [tex](y^2 + 2xy)(2at) + (2xy + x^2)(2a)[/tex]

Step 5: Replace x and y with their expressions in terms of t: x = at^2 and y = 2at. dz/dt = [tex]((2at)^2 + 2(at^2)(2at))(2at) + (2(at^2)(2at) + (at^2)^2)(2a)[/tex]

Step 6: Simplify the expression.

dz/dt = [tex](4a^2t^2 + 4a^2t^3)(2at) + (4a^2t^3 + a^4t^4)(2a)[/tex]

dz/dt = [tex]8a^3t^3 + 8a^3t^4 + 8a^3t^3 + 2a^5t^4[/tex]

dz/dt = [tex]16a^3t^3 + 10a^3t^4[/tex]

So, dz/dt for the given functions is [tex]16a^3t^3 + 10a^3t^4.[/tex]

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The value of k is 4 when volume of cylinder is [tex]\pi[/tex] .

To solve this problem, we need to use the formulas for the volumes of a cylinder and a cube.

The volume of a cylinder is given by V_cylinder = π[tex]r^{2}[/tex]h, where r is the radius and h is the height.

The volume of a cube is given by V_cube = [tex]s^{3}[/tex], where s is the length of a side.

In this problem, the cylinder just fits inside the cube, which means that the diameter of the cylinder is equal to the length of a side of the cube, or 2r = m. Therefore, the radius of the cylinder is m/2 cm, and the height of the cylinder is m cm.

Substituting these values into the formula for the volume of the cylinder, we get:

V_cylinder = π[tex](m/2)^{2}[/tex](m) = π[tex]m^{3/4}[/tex]

Substituting the value for the volume of the cylinder into the given ratio, we get:

k : π = V_cube : V_cylinder = [tex]m^{3}[/tex] : (π[tex]m^{3/4}[/tex] ) = 4 : π

Therefore, the value of k is 4.

Correct Question:

A cylinder just fits inside a hollow cube with sides of length m cm. The radius of the cylinder is m/2 cm. The height of the cylinder is m cm. The ratio of the volume of the cube to the volume of the cylinder is given by volume of cube : volume of cylinder = k : [tex]\pi[/tex], where k is a number. Find the value of k.

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To find the solution to the given differential equation y'' + 3y' + 2.25y = -10e^(-1.5x), you need to solve it using the following steps:

1. Identify the characteristic equation: r^2 + 3r + 2.25 = 0
2. Solve for r: r = -1.5, -1.5 (repeated root)
3. Find the complementary function (homogeneous solution): y_c(x) = C1 * e^(-1.5x) + C2 * x * e^(-1.5x)
4. Find a particular solution using an appropriate method, such as the method of undetermined coefficients: y_p(x) = A * e^(-1.5x)
5. Substitute y_p(x) into the given differential equation and solve for A: A = -10
6. Combine the complementary function and particular solution to find the general solution: y(x) = C1 * e^(-1.5x) + C2 * x * e^(-1.5x) - 10 * e^(-1.5x)

The general solution to the given differential equation is y(x) = C1 * e^(-1.5x) + C2 * x * e^(-1.5x) - 10 * e^(-1.5x).

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