Find the linearization of the function f(x,y) = √x^2 + 16y^2 at the point (3,1), and use it to approximate f(2.9.1.1)

The  correct choice is:

A(BC) =

-722 -164

-184 -72

We solve the matrix operation A(BC) as follows:

BC =

4(9) + (-2)(-2) + (-6)(-1) 4(8) + (-2)(-4) + (-6)(-2)

1(9) + 9(-2) + (-4)(-1) 1(8) + 9(-4) + (-4)(-2)

BC =

46 44

-15 -34

we then multiply A with BC as follows:

A(BC) =

-7 -8

-4 4

×

46 44

-15 -34

=

((-7)(46) + (-8)(-15)) ((-7)(44) + (-8)(-34))

((-4)(46) + (4)(-15)) ((-4)(44) + (4)(-34))

=

-722 -164

-184 -72

In a matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

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The volume of the solid of revolution is approximately 23.5619 cubic units.

To find the volume of the solid of revolution obtained by rotating the region bounded by y = r² and y = 1 about the horizontal line y = 1, you can use the disk method. Here's the step-by-step explanation:

1. First, determine the limits of integration. Since y = r², r = sqrt(y). The curve intersects y = 1 when r² = 1, so r = 1. The limits of integration are from 0 to 1.

2. Next, find the radius of each disk, which is the distance from the curve y = r² to the horizontal line y = 1. The radius is (1 - r²).

3. Now, find the area of each disk. The area is given by A(r) = π(radius)² = π(1 - r²)².

4. Finally, integrate the area function from 0 to 1 to find the volume of the solid of revolution: V = ∫[0,1] π(1 - r²)² dr.

Evaluating the integral, you get V ≈ 23.5619.

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Answer:

Step-by-step explanation:

easy its a

Household earning $55000 or more will act as larger potential customer base .

Given,

Earnings of household.

Here,

Earning are categorised on two incomes.

1st : Household earning $55000 or more will likely go to new shoe store .

2nd : Household earning $25000 or more will likely go to coffee shop .

Thus the household that earns more money will become potential customers for more number of things rather than household earnings less amount .

So, The households having more income will become larger potential customer .

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A corporation is creating a sinking fund to replace machinery in 3 years. In order to have $340,000 in the fund, they need to calculate how much to place in the account at the end of each month. Assuming an annual interest rate that is compounded monthly, the answer is $61,025 rounded to the nearest cent.

To calculate how much interest they would earn over the life of the account, we would need to know the interest rate.

To determine the value of the fund after 2, 4, and 6 years, we would need to know the interest rate and the amount placed in the account each month.

To calculate how much interest was earned during the second month of the 4th year, we would need to know the interest rate and the amount in the fund at that time.

A corporation creates a sinking fund to have $340,000 in 3 years for machinery replacement. The account's annual interest rate is compounded monthly. To determine the monthly deposit amount, we can use the sinking fund formula:

FV = PMT * (((1 + r)^nt - 1) / r)

where FV is the future value of the account ($340,000), PMT is the monthly deposit amount, r is the monthly interest rate, n is the number of times the interest is compounded per year (12 for monthly), and t is the number of years (3 in this case).

We need to solve for PMT:

$340,000 = PMT * (((1 + r)^36 - 1) / r)

To find the monthly deposit amount, we need the annual interest rate (not provided in the question). Once we have the interest rate, we can find the PMT value and calculate the interest earned over the account's life, as well as the fund's value after 2, 4, and 6 years. Additionally, we can determine the interest earned during the second month of the 4th year.

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We can say with 99% confidence that the proportion of all students at the college who will vote for Jennifer is between 0.729 and 0.797.

To construct a confidence interval for the proportion of all students at the college who will vote for Jennifer, we can use the following formula:

[tex]CI = p + z\times \sqrt{(p\times(1-p)/n)}[/tex]

where p is the sample proportion, z is the z-score for the desired confidence level, and n is the sample size.

First, we need to calculate the sample proportion:

p = 599/785 = 0.763

Next, we need to find the z-score for a 99% confidence level. From the standard normal distribution table, the z-score for a 99% confidence level is 2.576.

Now we can plug in the values and calculate the confidence interval:

[tex]CI = 0.763 + 2.576\times \sqrt{ (0.763\times (1-0.763)/785)}[/tex]

  = 0.763 ± 0.034

  = (0.729, 0.797)

Therefore, we can say with 99% confidence that the proportion of all students at the college who will vote for Jennifer is between 0.729 and 0.797.

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The mean of the distribution of sample means (also known as the expected value of the sample mean) is equal to the population mean, which is u = 118.9.

The standard deviation of the distribution of sample means (also known as the standard error of the mean) is equal to the population standard deviation divided by the square root of the sample size. Therefore,

o/sqrt(n) = 22.3/sqrt(94) = 2.30 (rounded to 2 decimal places)

So the standard deviation of the distribution of sample means is 2.30.

For a population with parameters μ = 118.9 (mean) and σ = 22.3 (standard deviation), if you draw a random sample of size n = 94, the mean of the distribution of sample means (us) is equal to the population mean, which is:

us = μ = 118.9

The standard deviation of the distribution of sample means, also known as the standard error, can be calculated using the formula:

Standard Error (SE) = σ / √n

In this case:

SE = 22.3 / √94 ≈ 2.30

So, the standard deviation of the distribution of sample means is approximately 2.30 (accurate to 2 decimal places).

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The value of the integral is 512/15. To evaluate the integral S4 0 (4-t)√t dt, we can use integration by substitution. Let u = √t, then du/dt = 1/(2√t), which implies that dt = 2u du.

Substituting u = √t and dt = 2u du, the integral becomes:

S4 0 (4-t)√t dt = S4 0 (4-t) u * 2u du = 2 S2 0 (4-u²) u² du

Now, we can expand the integrand and integrate term by term:

2 S2 0 (4-u²) u² du = 2 [∫4 0 u² du - ∫4 0 u⁴ du]

= 2 [(u³/3) |4 0 - (u⁵/5) |4 0]

= 2 [(64/3 - 64/5) - (0 - 0)]

= 2 [(320/15) - (64/15)]

= 2 [(256/15)]

= 512/15

Therefore, the value of the integral is 512/15.

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Answer:

the correct is D the graph is a decreasing function

The correct interpretation of the 95% confidence interval (-0.223, 0.988) for the mean yield difference between Silver Queen and Country Gentlemen corn varieties is that we can be 95% confident that the true mean difference in yield (Silver Queen - Country Gentlemen) falls within this range.

This means that, on average, Silver Queen could yield anywhere from 0.223 units less to 0.988 units more than Country Gentlemen with a 95% level of confidence.

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Answer:

8i + 34j

Step-by-step explanation:

[tex]4v - 2w \\ 4(5i + 4j) - 2(6i - 9j) \\ 20i + 16j - 12i + 18j \\ 20i - 12i + 16j + 18j \\ 8i + 34j[/tex]

The series diverges and the nth term test is used to identify the divergence.

It looks like there is a typo in the series.

The series should have a common difference between terms.

Assuming that the series is:

31 + 61 + 91 + ... + (30n+1)

We can use the nth term test to determine the convergence of the

series.

The nth term of the series is given by:

an = 30n + 1

As n goes to infinity, the dominant term in the nth term expression is

30n.

Therefore, the series diverges since the nth term does not approach

zero.

Hence, the series diverges and the nth term test is used to identify the

divergence.

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The difference between the linear and projectile motion is that "Linear-motion" refers to motion of object in a "straight-line", while "projectile-motion" refers to motion of an object that is thrown into air, in a curved path.

In linear motion, the object moves along a straight line, with its velocity and acceleration aligned in the same direction. The object's speed and direction may change, but its motion remains linear.

In projectile motion, the object moves along a curved path under the influence of gravity. The object is launched into the air with an initial velocity, and then gravity causes it to follow a parabolic path until it lands back on the ground. The motion of the object is influenced by both its initial velocity and the force of gravity.

In both linear and projectile motion, the "horizontal-component" of velocity will usually remain constant because there is no external force acting on the object in horizontal direction, and thus no acceleration.

Therefore, the object will continue to move at a constant velocity in the horizontal direction, as long as there is no external force acting on it.

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Using logarithmic differentiation the derivative of the function [tex]y'(x) = (ln(x))cos(6x) * [(-6sin(6x) * ln(ln(x))) + (cos(6x) * (1/x) * (1/ln(x)))][/tex]

Here's the function and the terms we'll be using:

Function:[tex]y = (ln(x))cos(6x)[/tex]

Terms: logarithmic differentiation, derivative

Step 1: Apply logarithmic differentiation by taking the natural logarithm (ln) of both sides of the equation.
[tex]ln(y) = ln((ln(x))cos(6x))[/tex]

Step 2: Simplify the right side of the equation using the properties of logarithms.
[tex]ln(y) = cos(6x) * ln(ln(x))[/tex]

Step 3: Differentiate both sides of the equation with respect to x using implicit differentiation.
[tex](d/dx) ln(y) = (d/dx) [cos(6x) * ln(ln(x))][/tex]

Step 4: Use the product rule on the right side of the equation. The product rule states that (uv)' = u'v + uv'.
[tex]y' / y = (-6sin(6x) * ln(ln(x))) + (cos(6x) * (1/x) * (1/ln(x)))[/tex]

Step 5: Multiply both sides of the equation by y to isolate y'.
[tex]y'(x) = y * [(-6sin(6x) * ln(ln(x))) + (cos(6x) * (1/x) * (1/ln(x)))][/tex]

Step 6: Substitute the original function y = (ln(x))cos(6x) back into the equation.
[tex]y'(x) = (ln(x))cos(6x) * [(-6sin(6x) * ln(ln(x))) + (cos(6x) * (1/x) * (1/ln(x)))][/tex]

That's your final answer for the derivative of the given function using logarithmic differentiation.

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The interval of convergence for the power series is (2-1/2, 2+1/2), or (3/2, 5/2).

Hence, the interval of convergence of the given power series is (3/2, 5/2).

To find the interval of convergence of the given power series:

We will first apply the Ratio Test:

[tex]|(-8)^{n+1} (x-2)^{3(n+1)} (n+1)^2 + 1| |(-8)^n (x-2)^{3n} (n^2 + 1)|[/tex]

___________________ = lim ____________________________________

[tex]|(-8)^n (x-2)^{3n} (n^2 + 1)| |(-8)^{n+1} (x-2)^{3(n+1)} (n+1)^2 + 1|[/tex]

Simplifying this expression, we get:

[tex]lim |(-8)(x-2)^3(n+1)^2 + 1|/(n^2+1)[/tex]

As n approaches infinity, the [tex](n+1)^2[/tex]  term in the numerator becomes dominant, so the limit simplifies to:

[tex]lim |-8(x-2)^3(n+1)^2|/n^2[/tex]

[tex]= 8|(x-2)|^3 lim (n+1)^2/n^2[/tex]

Using the limit properties, we can simplify the above limit to:

[tex]8|(x-2)|^3 lim (1 + 1/n)^2[/tex]

As n approaches infinity, the term [tex](1/n)^2[/tex] becomes negligible and the limit simplifies to:

[tex]8|(x-2)|^3 lim 1 = 8|(x-2)|^3[/tex]

Thus, the series converges absolutely if[tex]8|(x-2)|^3[/tex] < 1.

Solving the above inequality for x, we get:

[tex]|8(x-2)^3|[/tex] < 1

Taking the cube root of both sides, we get:

| x - 2 | < 1/2.

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The events A and B are not independent. This can be answered by the concept of Probability.

Two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event occurring. In this case, we are given that the probability of event A occurring is 0.2, the probability of event B occurring is 0.3, and the probability of both events occurring is 0.05.

To determine if the events A and B are independent, we can check if the probability of event A occurring is the same whether or not event B has occurred. Similarly, we can check if the probability of event B occurring is the same whether or not event A has occurred.

Let's start with the probability of event A occurring given that event B has occurred. We can use conditional probability to calculate this:

P(A|B) = P(A and B) / P(B)

Substituting the given probabilities, we get:

P(A|B) = 0.05 / 0.3 = 1/6

Now, let's compare this to the probability of event A occurring without any knowledge of event B, which is simply given as P(A) = 0.2. Since P(A|B) is not equal to P(A), we can conclude that event A is dependent on event B.

Similarly, we can check if event B is dependent on event A:

P(B|A) = P(A and B) / P(A)

Substituting the given probabilities, we get:

P(B|A) = 0.05 / 0.2 = 1/4

Again, since P(B|A) is not equal to P(B) = 0.3, we can conclude that event B is also dependent on event A.

Therefore, we can conclude that events A and B are not independent.

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The number of baseball teams of nine members that can be chosen from among twelve boys, without regard to the position played by each member is 220.

To solve this problem, we need to use the combination formula. The formula is:
nCr = n / r(n-r)
where n is the total number of items, r is the number of items we want to choose,
In this case, we have 12 boys and we want to choose a team of 9. So we have:
n = 12
r = 9
Plugging these values into the formula, we get:

12C9 = 12 / 9(12-9)
         = (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4) / (9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
         = 220
Therefore, there are 220 ways to choose a baseball team of nine members from among twelve boys, without regard to the position played by each member.

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An alternative method involves using the divisibility rule of 3, which states that a number is divisible by 3 if the sum of its digits is divisible by 3.

Since we are looking for multiples of 3, we can apply this rule to the digits of each number in the allowed set. We notice that the digits 1, 2, and 6 are already multiples of 3, so any combination of these digits will be a multiple of 3. Similarly, the digits 3 and 9 are also multiples of 3, but they are not in the allowed set. Therefore, any combination of the allowed digits that contains 3 or 9 will not be a multiple of 3.

Now, we consider the even numbers. An even number is a multiple of 2, so it must end in either 2 or 8.

Therefore, any combination of the allowed digits that ends in any other digit will not be even. Using this information, we can generate a list of all possible combinations of the allowed digits that end in 2 or 8.

Then, we check which ones have a sum of digits that is a multiple of 3. Finally, we eliminate any combination that repeats digits.

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Complete Question:

Ms. Hannabal assigned the students in her math class the task of finding all the multiples of 3 that even numbers and contain only the digits 1, 2, 3, 6, or 8. She told them not to repeat digits within a number, so a number like 222 would not be considered a solution

this was the result:

126, 128, 132, 136, 138, 162, 164, 168, 186, 212, 216, 218, 312, 316, 318, 362, 364, 368, 612, 614, 618, 812, 814, 816, 832, 834, 836

students identified 10 even three-digit multiples of 3, and these numbers are 126, 138, 162, 168, 186, 312, 318, 362, 368, and 618.

How could this be solved in a different way?

Answer:

S = 2π(7^2) + π(7)(15) = 637.42 square cm.

D is correct.

The exact surface area is 203π square cm., or about 637.74 square cm.

The normal vector of the tangent plane to the surface defined by (b) <-44e168, -80e168, 0>.

The normal vector of the tangent plane to a surface at a given point is perpendicular to the tangent plane and points outward from the surface. In this case, the surface is defined by the equation z = ex² + 18xy + 2y², and the point of interest is (-4, -2, e⁹⁶).

To find the normal vector, we need to calculate the gradient of the surface at the given point, which involves finding the partial derivatives of the surface equation with respect to x, y, and z, and evaluating them at the point (-4, -2, e⁹⁶).

The resulting vector will be the normal vector of the tangent plane at that point.

Taking the partial derivatives of the surface equation, we get:

∂z/∂x = 2ex² + 18y

∂z/∂y = 18x + 4y

Evaluating these partial derivatives at (-4, -2, e⁹⁶), we get:

∂z/∂x at (-4, -2, e⁹⁶) = 2e(-4)^2 + 18(-2) = -44e¹⁶⁸

∂z/∂y at (-4, -2, e⁹⁶) = 18(-4) + 4(-2) = -80e¹⁶⁸

Hence , the normal vector of the tangent plane at the point (-4, -2, e⁹⁶) is <-44e¹⁶⁸, -80e¹⁶⁸, 0>, which corresponds to option (b) <-44e168, -80e168, 0>.

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Our general solution is  y = c1e((-3 + √(13))/2)t + c2e((-3 - √(13))/2)t  where c1 and c2 are constants determined by initial or boundary conditions.

To find the general solution to this differential equation, we first need to find the characteristic equation by assuming that y = e(rt) and substituting it into the equation.

So we have y" + 3y' - y = 0
Substituting y = e(rt) we get
r² e(rt) + 3r e(rt) - e(rt) = 0

Dividing by e^(rt) we get
r² + 3r - 1 = 0

Now we solve for r by using the quadratic formula:
r = (-3 ± √(3² - 4(1)(-1))) / (2(1))
r = (-3 ± √(13)) / 2

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The equation of the line with value of a = 20 in the standard form ax + by  =c is equal to 20x - 5y = -10.

An equation of the line is,

ax + by = c

Equation is,

x + 4y = 19

Equation can be rearranged into the standard form,

⇒ x + 4y = 19

⇒4y = -x + 19

⇒y = (-1/4)x + (19/4)

Line is perpendicular to this line,

⇒ Slope is the negative reciprocal of (-1/4).

m = -1/m₁

   = -1/(-1/4)

   = 4

Since the line has y-intercept 2,

Use the point-slope form of the equation of a line

Then the equation of the line is,

y - y₁= m(x - x₁)

Substitute the values we have,

⇒ y - 2 = 4(x - 0)

⇒y - 2 = 4x

Rearranging this equation into the desired form ax + by = c, we get,

-4x + y =2

Multiplying both sides by -5 to ensure that a = 20

And there are no common factors between a, b, and c,

20x - 5y = -10

Therefore, the equation of the line in the desired form is 20x - 5y = -10.

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Step-by-step explanation:

The given line has slope , m = 4   parallel will be the same value m

  using the point (-1.6)   and slope m = 4

     y - 6 = 4(x -  -1)

(Which reduces to        y - 6 = 4 ( x+1) )

(a) The left-hand sum estimate for the integral is 1.214

(b) The right-hand sum estimate is 1.642, both rounded to three decimal places.

(a) To estimate the integral of eˣ using n=5 rectangles with left-hand sum and right-hand sum, we will first find the width of each rectangle (Δx) and then calculate the area of the rectangles using the function values.

Step 1: Calculate Δx
Δx = (b-a)/n = (1-0)/5 = 0.2

Step 2: Calculate the left-hand sum (LHS)
LHS = Δx * (f(x0) + f(x1) + f(x2) + f(x3) + f(x4))
LHS = 0.2 * ([tex]e^0+e^0^.^2+e^0^.^4+e^0^.^6+e^0^.^8[/tex])

LHS ≈ 1.214

(b) Step 3: Calculate the right-hand sum (RHS)
RHS = Δx * (f(x1) + f(x2) + f(x3) + f(x4) + f(x5))
RHS = 0.2 * ([tex]e^0+e^0^.^2+e^0^.^4+e^0^.^6+e^0^.^8[/tex])
RHS ≈ 1.642

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The correct answer is not (B). The critical value or values of X2 based on the given information are not -14.573 and 14.573. The correct answer is (C) 16.151, 40.113.

To find the critical value or values of X2, we need to refer to the Chi-Square distribution table or use a calculator that can calculate Chi-Square probabilities.

Given information:

Hypothesis: H1: σ ≠9.3 (which means the population standard deviation is not equal to 9.3)

Sample size: n = 28

Significance level: α = 0.05 (which corresponds to a 95% confidence level)

We need to find the critical value or values of X2 at a significance level of 0.05 with 27 degrees of freedom (n - 1 = 28 - 1 = 27) because we are dealing with a sample size of 28.

Using a Chi-Square distribution table or a calculator, the critical value of X2 at a significance level of 0.05 with 27 degrees of freedom is found to be 40.113. Since X2 is always positive, we only need to consider the upper tail of the Chi-Square distribution. Therefore, the critical value or values of X2 based on the given information are 16.151 and 40.113.

Therefore, the correct answer is (C) 16.151, 40.113.

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To find the total cost of producing 42 units, we will consider both the fixed costs and the variable costs. Since the marginal cost (MC) is given as $170.017 per unit, we can calculate the variable costs by multiplying MC by the number of units produced.

Variable costs = MC × Number of units = 170.017 × 42 = $7,140.714
Now, we'll add the fixed costs to the variable costs to get the total cost:
Total cost = Fixed costs + Variable costs = $3,350 + $7,140.714 = $10,490.714
Rounding to the nearest cent, the total cost of producing 42 units is $10,490.71.

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Answer: 180

Step-by-step explanation: 1440 divided by 8 is 180.

The formula for f(t) is calculated to be f(t) = 15e^(-0.1714t)

Let's start with the given information: the rate of change of f(t) is proportional to the value of f(t) at any time. This means that we can write:

f'(t) = k*f(t)

where k is the proportionality constant. To solve for f(t), we need to find the value of k.

We know that 15 units of medicine were injected initially, and after 35 minutes, only 9 units remained. Let's use this information to find k.

We can write the following equation to represent the rate of change of f(t):

f'(t) = -r*f(t)

where r is the rate at which the medicine is leaving the dog's body. We know that after 35 minutes, 6 units of medicine were used, so:

r = (6 units) / (35 minutes) = 0.1714 units/minute

Now we can solve for k by using the given information that at t=0, f(0) = 15:

f'(t) = k*f(t)

f'(0) = kf(0) = -rf(0)

k = -r = -0.1714

So now we have k and we can solve for f(t) using the differential equation:

f'(t) = -0.1714*f(t)

Separating variables and integrating, we get:

ln(f(t)) = -0.1714*t + C

where C is the constant of integration. Solving for f(t), we get:

f(t) = e^(-0.1714*t + C)

To find the value of C, we use the initial condition f(0) = 15:

f(0) = e^(C) = 15

C = ln(15)

So the final formula for f(t) is:

f(t) = 15e^(-0.1714t)

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The total weight of the container's contents is 11 pounds.

The container's volume can be estimated by multiplying the length, breadth, and height:

2 feet * 2 feet * 12.5 feet equals 50 cubic feet

Because the contents weigh 0.22 pounds per cubic foot, calculating the volume by the weight per cubic foot yields the total weight of the contents:

50 cubic feet * 0.22 pounds per cubic foot = 11 pounds

As a result, the total weight of the container's contents is 11 pounds.

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Answer:

12.4 ft

Step-by-step explanation:

a² + b² = c²

4² + h² = 13²

16 + h² = 169

h² = 169 - 16

h² = 153

h = 12.369316

Answer: 12.4 ft

Answer:

12.4 ft

Step-by-step explanation:

4² + h² = 13²

h² = 13² - 4² = 169 - 16 = 153

h = √153 ≈ 12.37 ft ≈ 12.4 ft