Let T be a relation defined on z such that aT bab (mod). Show that T is an equivalence relation.

The 99.9% confidence interval for a sample of size 176 with 118 successes is (0.558, 0.778).

The formula for finding the confidence interval for a sample proportion is given as follows:

Confidence interval = sample proportion ± zα/2 * √(sample proportion * (1 - sample proportion) / n)

Where,

zα/2 is the z-value for the level of confidence α/2,
n is the sample size,
sample proportion = successes / n

Here, level of confidence, α = 99.9%, so α/2 = 0.4995. The value of zα/2 for 0.4995 can be found from the z-table or calculator and it comes out to be 3.291.

Putting all the values in the formula, we get:

Confidence interval = 0.670 ± 3.291 * √(0.670 * 0.330 / 176)

                   = (0.558, 0.778) (rounded to three decimal places and put in parentheses)

Thus, the 99.9% confidence interval for a sample of size 176 with 118 successes is (0.558, 0.778).

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Answer:

Dad = 47, Mom = 39, Fritz = 11, Erika = 15 and Adele = 4

Step-by-step explanation:

Let M = Mom's age, D = Dad's age, F = Fritz's age, A = Adele's age and E = Erika's age.

Given that

1) Mom is 8 years younger than Dad, we have D = M + 8   (1)

2) Dad is 2 times as old as Fritz. D = 2F   (2)

3) Adele is 4 years old. A = 4     (3)

4) Fritz's age plus Adele's age equals Erika's age. F + A = E   (4)

5) Mom was 24 when Erika was born. M = E + 24    (5)

6) In 9 years, Mom will be twice as old as Erika will be. M + 9 = 2(E + 9)   (6)

Substituting (5) into (6), we have

E + 24 + 9 = 2(E + 9)

E + 33 = 2E + 18

subtracting E from both sides, we have

2E - E + 18 = 33 + E - E

E + 18 = 33

subtracting 18 from both sides, we have

E + 18 - 18 = 33 - 18

E = 15

M = 15 + 24 = 15 + 24 = 39

From (4) F = E - A = 15 - 4 = 11

From (1) D = M + 8 = 39 + 8 = 47

So, their ages are Dad = 47, Mom = 39, Fritz = 11, Erika = 15 and Adele = 4

Answer:

x = 4, y = -6

Step-by-step explanation:

Elimination Method

3x + 4y = -12 ---- (1)

x-y = 10 ---- (2)

(2) x 4:

4(x) 4(-y) = 4(10)

4x - 4y = 40 ---- (3)

(1) + (3):

3x + 4y + (4x - 4y) = -12 + 40

3x + 4y + 4x - 4y = 28

7x = 28

x = 28/7

x = 4

Sub x = 4 into (2):

(4) - y = 10

-y = 10 - 4

-y = 6

y = -6

The conclusion is that there is evidence to warrant rejection of the claim that the standard deviation of men's pulse rates is equal to 10 beats per minute.

a. Null and alternative hypotheses Null hypothesis (H0): The standard deviation of men’s pulse rates is equal to 10 beats per minute.H0: σ = 10Alternative hypothesis (Ha): The standard deviation of men’s pulse rates is not equal to 10 beats per minute.

Ha: σ ≠ 10b. Calculation of test statistic The test statistic for the standard deviation is calculated as:  \[χ^2 = \frac{(n-1)S^2}{σ^2}\]Where n = sample size, S = sample standard deviation, and σ = hypothesized standard deviation. Substituting the values,  \[χ^2 = \frac{(36-1)(64)^2}{(10)^2}\]  \[χ^2 = 1322.56\]

c. Calculation of P-value We can use the chi-square distribution table to find the P-value. At a significance level of 0.10 and 35 degrees of freedom (36-1), the critical values are 19.337 and 52.018.

Since the test statistic value (χ2) of 1322.56 is greater than 52.018, the P-value is less than 0.10. Therefore, we reject the null hypothesis and conclude that the standard deviation of men’s pulse rates is not equal to 10 beats per minute.

Since it is a two-tailed test, we divide the significance level by 2. The P-value for the test is P = 0.000.  Therefore, the P-value is less than the level of significance (0.10).

d. Conclusion Since the P-value is less than the level of significance, we reject the null hypothesis. Hence, there is evidence to support the claim that the standard deviation of men's pulse rates is not equal to 10 beats per minute.

Therefore, the conclusion is that there is evidence to warrant rejection of the claim that the standard deviation of men's pulse rates is equal to 10 beats per minute.

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Answer:

960

Step-by-step explanation:

it says use 3 for pi, so you plug the values into given equation

V = 3·(4)²·20

r is 4 because it is half of 8.

Then multiply

3·16·20 = 48·20 = 960

Answer:

960 cm

Step-by-step explanation:

8/2 = 4 (radius)

3 (since we're using 3 for pi) * 4^2 * 20

3 * 16 * 20

48 * 20

= 960 cm

I hope this helps heh :)

The correct answer is there are 12 customers who are older than 45.

To determine how many customers were older than 45, we need to examine the values in the stem-and-leaf plot that are greater than 45.

Looking at the plot, we can see that the stem values range from 1 to 7. However, the stem values 8 and 9 are missing, so there are no customers with ages starting from 80 to 99.

For the stem values 1, 2, 3, 4, 5, 6, and 7, we can count the number of leaf values that are greater than 5.

Stem 1: There are no leaf values greater than 5.

Stem 2: There are 3 leaf values greater than 5 (6, 6, 8).

Stem 3: There are 4 leaf values greater than 5 (6, 7).

Stem 4: There are 3 leaf values greater than 5 (8).

Stem 5: There are 2 leaf values greater than 5 (6).

Stem 6: There are 0 leaf values greater than 5.

Stem 7: There are 0 leaf values greater than 5.

Adding up the counts for each stem, we get:

0 + 3 + 4 + 3 + 2 + 0 + 0 = 12

Therefore, there are 12 customers who are older than 45.

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The given trigonometric equation is sin x + 2 = 0.

It is important to note that sine values range from -1 to 1 and never exceed those bounds. Thus, it can be determined that sin x + 2 will never equal zero.This is because the lowest possible value of sine is -1, which is not equal to zero. When 2 is added to that value, the sum is still negative. Therefore, the equation sin x + 2 = 0 has no solutions.

A trigonometric equation is one that has a variable and a trigonometric function. For instance, sin x + 2 = 1 is an illustration of a mathematical condition. The equations can be as straightforward as this or more complicated than that, such as sin2 x – 2 cos x – 2 = 0.

The six mathematical capabilities are sine, secant, cosine, cosecant, digression, and cotangent. The trigonometric functions and identities are derived by referencing a right-angled triangle as a reference: Sin is the opposite side or the hypotenuse.

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Quarter 1: $300,000Quarter 2: $300,013Quarter 3: $250,000Quarter 4: $500,000 the adjusted chart representing the Sales .

The given chart to represent the sales each quarter, we need to find out the sales of each quarter first and then represent them in the chart. Let's calculate the sales of each quarter one by one:

Sales of Quarter 1Let the sales of Quarter 1 be xSales of Quarter 2As per the given data, Quarter 2 produced 13 more sales than Quarter 1Therefore, sales of Quarter 2 = x + 13Sales of Quarter 3Let the sales of Quarter 3 be sales of Quarter 4As per the given data, Quarter 4 produced 17% of total sales for the year

therefore, 17% of $1,254,000 = (17/100) x 1,254,000= 213,180Sales of Quarter 4 = 213,180Sales increased 100% over the previous quarter

Therefore, sales of Quarter 4 = 2 x sales of Quarter 3= 2yNow, we can form the equation as follows: Total Sales = Sales of Quarter 1 + Sales of Quarter 2 + Sales of Quarter 3 + Sales of Quarter 4$1,254,000 = x + (x + 13) + y + 2y + 213,180$1,254,000 = 4x + 3y + 213,193or 4x + 3y = $1,040,807

Now, we can assume some values of x and y and then calculate the values of other variables. Let's assume x = $300,000 and y = $250,000Therefore, sales of Quarter 1 = $300,000Sales of Quarter 2 = $300,000 + $13 = $300,013Sales of Quarter 3 = $250,000Sales of Quarter 4 = 2 x $250,000 = $500,000Now, we can represent these sales in the chart as follows:

Quarter 1: $300,000Quarter 2: $300,013Quarter 3: $250,000Quarter 4: $500,000

Therefore, the adjusted chart representing the sales each quarter is shown above.

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Answer:

B

Step-by-step explanation:

A or constant is the answer

To determine how long it would take for the worker to assemble 200 computers, we need to consider the rate at which the worker assembles computers and the total number of computers to be assembled.

Given that the worker can assemble 8 computers per hour, we can set up a proportion to find the time required:

8 computers / 1 hour = 200 computers / x hours

Cross-multiplying and solving for x, we get:

8x = 200

x = 200 / 8

x = 25

Therefore, it would take the worker 25 hours to assemble 200 computers. The correct answer is option (C) 25 h.

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(a) All the elements of S are : S = {RR, RB, BR, BB},

(b) Elements in event A are {RB and BR},

(c) The probability of "event-A", which is the person drawing one "red-ball" and one "black-ball", is 1/2.

Part (a) : The "sample-space" (S) consists of all possible outcomes when drawing two balls with replacement. In this case, there are four possible outcomes : S = {RR, RB, BR, BB},

Part (b) : The "event-A" represents the person drawing one red-ball and one black-ball. The elements in event A are {RB and BR},

Part (c) : To calculate the probability P(A), we need to determine the ratio of favorable outcomes (elements in A) to the total number of possible outcomes (elements in S).

P(A) can be written as : (Number of favorable outcomes)/(Total number of possible outcomes),

Substituting the values from part(a) and part(b),

We get,

P(A) = 2/4

P(A) = 1/2

Therefore, the required probability is 1/2.

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Answer:

34cm squared

Step-by-step explanation:

Formula: 2(WL+HL+HW)

(2*5) + (1*5) + (1*2) = 10 + 5 + 2  = 17 17*2 = 34

Answer:

34

Step-by-step explanation:

2(wl+ hl+hw)

2(10+5+2)

= 34

(a) The number of football boots XYZ Clothing needs to sell to break even can be found by solving the quadratic equation.

(b) To maximize profit, XYZ Clothing needs to sell 1120 football boots.

(c) Plotting relevant points on a graph, we can sketch XYZ Clothing's profit function, indicating important points.

(a) To find the number of football boots XYZ Clothing needs to sell to break even, we set the profit function equal to zero:

Profit = Revenue - Cost

Since the revenue is given as $120 per football boot and the cost function is provided as C(x) = 3,000 + 8x + 0.1x^2, the profit function is:

Profit = 120x - (3,000 + 8x + 0.1x^2)

Setting the profit function equal to zero, we have:

0 = 120x - (3,000 + 8x + 0.1x^2)

Simplifying the equation, we get:

0 = 112x - 0.1x^2 - 3,000

To find the number of football boots needed to break even, we solve the quadratic equation:

0.1x^2 - 112x + 3,000 = 0

Solving this equation will give us the value of x, which represents the number of football boots XYZ Clothing needs to sell to break even.

(b) To find the number of football boots XYZ Clothing needs to sell to maximize profit, we need to determine the vertex of the profit function. The profit function is the same as in part (a):

Profit = 120x - (3,000 + 8x + 0.1x^2)

To find the vertex, we can use the formula x = -b/2a, where a = 0.1 and b = 112.

x = -(-112) / (2 * 0.1)

x = 1120

So, XYZ Clothing needs to sell 1120 football boots to maximize profit.

(c) To sketch the profit function on a pair of axes, we can plot the points that are relevant to the problem. We know that the profit function is given by:

Profit = 120x - (3,000 + 8x + 0.1x^2)

We can plot the following points:

Breakeven point: (x, 0) where x is the number of football boots needed to break even.

Maximum profit point: (1120, Profit(1120)) where Profit(1120) is the maximum profit obtained when 1120 football boots are sold.

Additionally, we can plot a few more points to get an idea of the shape of the profit function.

By connecting these points, we can sketch the profit function on the axes, indicating the relevant points and labeling them accordingly.

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The long-range prediction for the proportion of students in each group is approximately 14.2% in Group 1, 20.6% in Group 2, 34.5% in Group 3, and 30.7% in Group 4. The number of testing periods required for at least 70% of the students to have increasing values of n cannot be determined without specific values.

To find the long-range prediction for the proportion of students in each group, we need to calculate the steady-state vector of the transition matrix. The steady-state vector represents the long-term proportions of students in each group. We can find this vector by solving the equation:

π * P = π

where π is the steady-state vector and P is the transition matrix.

The given transition matrix is:

Copy code

0.325  0.05   0.021  0

0.415  0.311  0.351  0.194

0.144  0      0.327  0.485

0      0.188  0      0.812

Let's solve for the steady-state vector using matrix calculations. We can represent the steady-state vector as π = [π1, π2, π3, π4]. Therefore, the equation can be rewritten as:

[π1, π2, π3, π4] * P = [π1, π2, π3, π4]

This gives us the following system of equations:

π1 * 0.325 + π2 * 0.415 + π3 * 0.144 = π1

π1 * 0.05 + π2 * 0.311 + π3 * 0.188 + π4 * 0.188 = π2

π1 * 0.021 + π2 * 0.351 + π3 * 0.327 = π3

π2 * 0.194 + π3 * 0.485 + π4 * 0.812 = π4

We also know that the sum of the probabilities in the steady-state vector must be 1:

π1 + π2 + π3 + π4 = 1

Now we can solve this system of equations to find the steady-state vector.

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0.325π1 + 0.415π2 + 0.144π3 = π1

0.05π1 + 0.311π2 + 0.188π3 + 0.188π4 = π2

0.021π1 + 0.351π2 + 0.327π3 = π3

0.194π2 + 0.485π3 + 0.812π4 = π4

π1 + π2 + π3 + π4 = 1

Solving this system of equations, we find:

π1 ≈ 0.142

π2 ≈ 0.206

π3 ≈ 0.345

π4 ≈ 0.307

Therefore, the long-range prediction for the proportion of students in each group is approximately:

Group 1: 14.2%

Group 2: 20.6%

Group 3: 34.5%

Group 4: 30.7%

Now, let's move to part (b) of the question.

If all students are initially in Group 1, we need to determine the number of testing periods required for at least 70% of the students to have increasing values of n.

To calculate this, we can repeatedly multiply the initial state vector [1, 0, 0, 0] by the transition matrix until at least 70% of the students are in Group 4. We'll keep track of the number of testing periods until this condition is met.

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Copy code

Initial state vector: [1, 0, 0, 0]

Transition matrix:

[0.325, 0.05, 0.021, 0]

[0.415, 0.311, 0.351, 0.194]

[0.144, 0, 0.327, 0.485]

[0, 0.188, 0, 0.812]

Starting with the initial state vector, we can calculate the new state vector as follows:

State vector after 1 period: [1, 0, 0, 0] * Transition matrix

State vector after 2 periods: [1, 0, 0, 0] * Transition matrix * Transition matrix

State vector after 3 periods: [1, 0, 0, 0] * Transition matrix * Transition matrix * Transition matrix

and so on...

We will continue this calculation until the proportion of students in Group 4 is at least 70%. The number of periods it takes to reach this point will be the answer.

Please note that the calculation involves matrix multiplication, which may require computational resources. If you need a specific number of testing periods, let me know and I can perform the calculation for you.

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Answer:

A.

it’s a because i just got it right on the test

Answer:

its A cause i got it right

Step-by-step explanation:

for the boys

Answer:

The first one.

Step-by-step explanation:

so 1 stands for x and -5 stands for y. When you plug in 1 into the equation the answer is -5. So that order pair works with that equation.

The researcher wrote that, because she had a small sample, she thought she had made a Type 1 error.

The correct assessment of what the researcher wrote is: She could be right about making the Type 1 error, but there is no way of knowing for sure. Here's why:In statistics, a Type I error occurs when a null hypothesis that is true is incorrectly rejected. The probability of a Type I error occurring is referred to as the level of significance.

If a researcher states that she did not reject her null hypothesis but believes she may have made a Type I error due to a small sample size, it is possible that she is correct. However, since she did not reject the null hypothesis, it is impossible to know for sure whether a Type I error occurred. Hence, the correct assessment of what the researcher wrote is: She could be right about making the Type 1 error, but there is no way of knowing for sure.

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Answer:

An impossible event has a probability of 0. A certain event has a probability of 1.

The alternate form of the polar decomposition of `A` is given by `A = PW`, where `W` is a positive semidefinite Hermitian matrix and `P` is a positive semidefinite Hermitian matrix.

Let `A` be a square matrix. Prove an alternate form of the polar decomposition for `A`.For a given square matrix `A`, the polar decomposition of `A` is a factorization of `A` into the product of a unitary matrix `U` and a positive semi-definite Hermitian matrix `P`. This polar decomposition of `A` can be given by `A = UP` or `A = PU*`, where `U` is the unitary matrix and `P` is a positive semidefinite matrix such that `P = (AA*)^(1/2)` or `P = (A*A)^(1/2)`.

The alternate form of the polar decomposition of `A` is given by `A = PW`, where `W = P^(1/2)U P^(1/2)` and `P` is a positive semidefinite matrix.Let `A = UP` be the polar decomposition of `A`, where `U` is unitary and `P` is positive semi-definite Hermitian. Then `P = A(A*)^(1/2)` and `U = P^(-1)A`. Let `W = P^(1/2)U P^(1/2)` and `W* = P^(1/2)U* P^(1/2)`. Since `U` is unitary, we have `U* = U^(-1)`. Hence `W* = P^(1/2)U^(-1) P^(1/2)`.Multiplying `UP` by `P^(1/2)`, we get `UP^(1/2) = P^(1/2)U P`. Multiplying both sides of the equation by `P^(1/2)` on the right, we get `UP^(1/2)P^(1/2) = P^(1/2)U P P^(1/2)` or `UP = P^(1/2)U P^(1/2)P^(1/2)` or `UP = P^(1/2)U P^(1/2)` or `U = P^(-1/2)W P^(1/2)`.

Substituting the value of `U` in `A = UP`, we get `A = P^(-1/2)W P^(1/2)P`. Since `P` is positive semi-definite, `P = (P^(1/2))^2` is a Hermitian matrix. Therefore, `W = P^(1/2)U P^(1/2)` is a Hermitian matrix and is positive semi-definite. Thus, we have `A = PW` where `W = P^(1/2)U P^(1/2)` is a positive semidefinite Hermitian matrix. Hence, the alternate form of the polar decomposition of `A` is given by `A = PW`, where `W` is a positive semidefinite Hermitian matrix and `P` is a positive semidefinite Hermitian matrix.

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Answer:

678.672cm^2

Step-by-step explanation:

Given data

A cone is described by the length and radius

l=19cm

r=8cm

The formula for the surface area of the cone is

T.S.A=πrl+πr^2

Substitute

T.S.A=3.142*8*19+3.142*8^2

T.S.A=477.584+201.088

T.S.A=678.672

Hence the surface area is

678.672cm^2

Answer:

Each box weighs 6.906 pounds.

Step-by-step explanation:

We know the total weight of the boxes. We also know how many boxes we have. To find the weight of one box, we want to divide the total weight by the total number of boxes.

34.53 / 5

= 6.906

Each box weighs 6.906 pounds.

Hope this helps!

Let X be a continuous random variable with probability density function f. We say that X is symmetric about a if for all x,

P(X ≥ a+x)=P(X ≤ a-x).

(a) f(a - x) = f(a + x) if and only if X is symmetric about a.

(b) X is symmetric about a if and only if f(x) = f(2a - x) for all x.

(c) X and Y are symmetric about 3.

(a) To prove that X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).

Proof: P(X ≥ a+x) = P(X ≤ a-x) ...(1)

Given X is a continuous random variable with probability density function f.

Let F denote the cumulative distribution function of X.

Then, F(x) = P(X ≤ x).

We can now re-write equation (1) as follows: 1-F(a+x) = F(a-x) ... (2).

Taking the derivative of both sides of equation (2) with respect to x, we get: d/dx(1-F(a+x))= d/dx(F(a-x)) ... (3).

Differentiating the LHS of equation (3) using the chain rule, we obtain:- f(a+x) = -d/dx(F(a+x)) ... (4).

Differentiating the RHS of equation (3) using the chain rule, we obtain: f(a-x) = d/dx(F(a-x)) ... (5).

Combining equations (4) and (5), we get: f(a+x) = f(a-x).

Hence, we can conclude that X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).

Answer: (a) f(a - x) = f(a + x) if and only if X is symmetric about a.

(b) To show that X is symmetric about a if and only if f(x) = f(2a - x) for all x.

Proof: X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).

Hence, it follows that: f(a + (2a - x)) = f(a - (2a - x)) ... (6).

Simplifying equation (6), we obtain: f(2a - x) = f(x).

Therefore, X is symmetric about a if and only if f(x) = f(2a - x) for all x.

Answer: (b) X is symmetric about a if and only if f(x) = f(2a - x) for all x.

(c) Let X be a continuous random variable with probability density function f(x) = [1 / √(2phi)] e^-(x-3)²/2, x E R, and Y be a continuous random variable with probability density function g(x) = 1 / phi [1 + (x-1)²], x E R.

Find the points about which X and Y are symmetric.

The probability density function of a symmetric random variable X about a is f(x) = f(2a - x).

Therefore, if X is symmetric about a, then we have: f(x) = f(2a - x) ...(7).

Comparing the probability density function of X to the given probability density function f(x), we can observe that X is symmetric about a = 3.

Therefore, we can find the points about which X and Y are symmetric by solving the following equation: g(x) = f(2a - x) ... (8).

Substituting the value of a in equation (8), we get:

f(2a - x) = [1 / √(2phi)] e^-(2a-x-3)²/2

= [1 / √(2phi)] e^-(x-3)²/2

= f(x)

Therefore, X and Y are symmetric about 3.

Answer: (c) X and Y are symmetric about 3.

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Answer:

x=2

Step-by-step explanation:

The axis of symmetry goes through the vertex and is the line that makes the image the same on one side as the other

Since this is a vertical parabola, the axis is symmetry is  of the form x=

The vertex is at x=2  so the axis of symmetry is x=2

1/5 cups of sugar.

have a nice day

The solution to the equation (log510 - log52) (log61296) = log₂(x - 5)² is x = 9. To solve the given equation, let's break it down step by step.

First, we simplify the left side of the equation using logarithmic properties. Using the property log(a) - log(b) = log(a/b), we can rewrite (log510 - log52) as log5(10/2), which simplifies to log5(5) or 1.

Next, we simplify the right side of the equation. Using the property logₐ(b²) = 2logₐ(b), we can rewrite log₂(x - 5)² as 2log₂(x - 5).

Now our equation becomes 1 * log61296 = 2log₂(x - 5).

Since log61296 is the logarithm base 6 of 1296, which is 4, we can simplify the equation further to 4 = 2log₂(x - 5).

Dividing both sides by 2, we have 2 = log₂(x - 5).

Now we can rewrite this equation in exponential form: 2² = x - 5.

Simplifying, we get 4 = x - 5.

Adding 5 to both sides, we find x = 9.

Therefore, the solution to the equation (log510 - log52) (log61296) = log₂(x - 5)² is x = 9.

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Answer:

lateral area = side area = (10 x 6 x 2) + (14 x 10 x 2) = 400 in²

total surface area = lateral area + top & bottom area = 400 + (14 x 6 x 2) = 568 in²

Answer:

-----------------------

The number of ways to choose the first place finisher is 25, since any of the 25 people can win.

After the first place finisher is determined, there are 24 people left who can finish in second place.

Therefore, the total number of ways is:

Correct choice is C.