Match the formulas for volume and calculate the volumes of the sphere, cylinder, and cone shown below. Each shape has a radius of 2.5 and the cylinder and cone have a height of 4.

options for each drop down box [choose]:
Sphere - volume measure
Sphere - volume formula
Cone - volume formula
Cone - volume measure
Cylinder - volume measure
Cylinder - volume formula
None of these options

The equation of the line is P = 2(1.0048)ᵗ and the population in 30 years is 2.31

The equation is represented as

y = abᵗ

Where

a = y when t = 0

The points on the line are

(0, 2) and (20, 2.2)

This means that

a = 2

So, we have

y = 2bᵗ

Using the points, we have

2b²⁰ = 2.2

b²⁰ = 1.1

So, we have

b = 1.0048

This means that the equation is

P = 2(1.0048)ᵗ

Above, we have

a = 2

So, the meaning of the interpretation is that the initial population of the endangered colony is 2

Also, we have

b = 1.0048

So, the meaning of the interpretation is that the endangered colony increases by a factor of 1.0048 every year

Recall that

P = 2(1.0048)ᵗ

Here, we have

t = 30

So, the equation becomes

P = 2(1.0048)³⁰

Evaluate

P = 2.31

Hence, the population in 30 years is 2.31

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Answer:

12.8

Step-by-step explanation:

First, we can simplify each fraction separately:

64e^2/5e = 64/5e^(1-1) = 64/5

3e/8e = 3/8

Now we can multiply:

(64/5) * (3/8) = 12.8

Therefore, the product is 12.8.

The equation that could represent each of the graphed polynomial function include the following:

First graph: y = x(x + 2)(x - 3)

Second graph: y = x⁴ - 5x² + 4

In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.

Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph.

By critically observing the graph of the polynomial function shown in the image attached above, we can logically deduce that the first graph has a zero of multiplicity 1 at x = 2 and zero of multiplicity 1 at x = -3.

Similarly, we can logically deduce that the second graph has a zero of multiplicity 2 at x = 2 and zero of multiplicity 2 at x = -2.

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The angle ZFD = 57°

Given that:

arc FZ = 66"

FD is the diameter ·Because passing through, Centre.

Angle formed arc FZ at centre = 66°

Angle formed the arc FZ at Circumference at Circle ¹/₂ x66 = 33°

FD is diameter

∠Z= 90°.

∠ZFD = ∠ in Δ FZD

∠ZFD + ∠FZD + ∠FDZ = 180°

∠ZFD + 90° + 33° = 180°

∠ZFD = 180° - 90° - 33° = 57°

Hence the ∠ZFD = 57°

Therefore, angle ZFD is 57°

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In this scenario, the dependent variable is "how many jumps I can do," while the independent variable is "how long I am jumping (2 minutes)."

The relationship between these variables can be described as follows: The number of jumps completed depends on the duration of time spent jumping, with a specific focus on a 2-minute interval.

When we say the dependent variable is "how many jumps I can do," it means that the number of jumps completed is determined by or depends on the independent variable, which is the duration of time spent jumping.

This suggests that as the duration of time increases or decreases, it will likely have an impact on the number of jumps performed.

In this particular case, you have specified a 2-minute interval as the focus. It suggests that you are examining the relationship between the number of jumps completed and the specific duration of 2 minutes.

This implies that you are interested in understanding how the number of jumps varies within this fixed time frame.

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The combination of side lengths that would not form a triangle is C.XY = 11 mm, YZ = 14 mm, XZ = 21 mm.

We shall use the triangle inequality theorem to determine if a set of side lengths can form a triangle.

The triangle inequality theorem says that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We shall calculate each of the options:

For option A:

XY + YZ = 7 mm + 14 mm = 21 mm which is < XZ = 25 mm.

Therefore, option A does form a triangle.

For option B:

XY + YZ = 11 mm + 18 mm = 29 mm, which is > XZ = 21 mm.

YZ + XZ = 18 mm + 21 mm = 39 mm, which is > XY = 11 mm.

XY + XZ = 11 mm + 21 mm = 32 mm, which is > YZ = 18 mm.

Therefore, option B does form a triangle.

For option C:

XY + YZ = 11 mm + 14 mm = 25 mm, and is > XZ = 21 mm.

Therefore, option C does not form a triangle.

For option D:

XY + YZ = 7 mm + 14 mm = 21 mm, which is > XZ = 17 mm.

YZ + XZ = 14 mm + 17 mm = 31 mm, which is > XY = 7 mm.

XY + XZ = 7 mm + 17 mm = 24 mm, which is > YZ = 14 mm.

Therefore, option D does form a triangle.

Therefore, the combination of side lengths that would not form a triangle is XY = 11 mm, YZ = 14 mm, XZ = 21 mm.

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The volume of the cone with a height and radius both of 7 units is 359.24 cubic units.

In Mathematics and Geometry, the volume of a cone can be determined by using this formula:

V = 1/3 × πr²h

Where:

By substituting the given parameters into the formula for the volume of a cone, we have the following;

Volume of cone, V = 1/3 × 3.142 × 7² × 7

Volume of cone, V = 1/3 × 3.142 × 49 × 7

Volume of cone, V = 359.24 cubic units.

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Answer:

343/3

Proof of answer is in the image, please give brainliest

The toys approach each other at the distance of  630 cm.

To solve the problem, we can use the Pythagorean theorem.

Let the distance between the tugboat and the sail ship be d, and

let t be the time in seconds since they started moving.

Then we have:

Distance traveled by the tugboat (in cm) = 5t

Distance traveled by the sail ship (in cm) = 7t/sqrt(2)

Using the Pythagorean theorem, we have:

d² = (5t)² + (7t/(\sqrt(2)))²

d² = 25t² + 24.5t²

d² = 49.5t²

d = \sqrt(49.5)t

To find how closely the two toys approach each other, we need to find the minimum value of d.

This occurs when t is maximized, which happens when the toys are closest to each other.

The sail ship travels a distance of 8\sqrt(2) meters in the Northeast direction, which is equivalent to 800\sqrt(2) cm. Therefore, the time taken for the sail ship to travel this distance is:

t = (800\sqrt(2) cm) / (7 cm/(\sqrt(2))) = 200\sqrt(2) seconds

Substituting this value of t in the equation for d, we get:

d = \sqrt(49.5)(200\sqrt(2)) = 630 cm (corrected)

Therefore, the minimum distance between the two toys is 630 cm.

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To find the value of x when (fog)(x) = -8, we first need to find the composition of f and g, which is given by (fog)(x) = f(g(x)). To do this, we substitute g(x) into the expression for f(x) and simplify:

f(g(x)) = f(1) when g(x) = 1

f(g(x)) = f(-2) when g(x) = -2

f(g(x)) = f(3) when g(x) = 3

f(g(x)) = f(-4) when g(x) = -4

f(g(x)) = f(4) when g(x) = 4

There is no value of x for which (fog)(x) = -8.

Using the table given in the question, we can find the values of f(g(x)) for each possible value of g(x):

f(g(x)) = f(1) = -2

f(g(x)) = f(-2) = 0

f(g(x)) = f(3) = 32

f(g(x)) = f(-4) = 8

f(g(x)) = f(4) = 4

Therefore, (fog)(x) = -8 is not possible. The closest value we can get to -8 is by setting g(x) = -4, which gives f(g(x)) = f(-4) = 8. Thus, there is no value of x for which (fog)(x) = -8.

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The missing side length is 5√3 centimeters.

We can use the Pythagorean theorem to find the missing side length. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. In equation form, this looks like:

a² + b² = c²

where a and b are the lengths of the legs, and c is the length of the hypotenuse.

To use this formula to solve for the missing side length, we can plug in the values we know:

5² + b² = 10²

We can simplify this equation by squaring 5 and 10:

25 + b² = 100

Next, we can isolate the variable (b) on one side of the equation by subtracting 25 from both sides:

b² = 75

Finally, we can solve for b by taking the square root of both sides:

b = √(75)

This simplifies to:

b = 5*√(3)

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Complete Question:

By using the Pythagoras theorem, Find the value of the Other side when the value of hypotenuse is 10 cm and the value of the side is 5 cm.

Answer:

The approximated measure of this angle is 90°, so this may be a right angle.

Place a sheet of paper so that the corner corresponds to the angle. You will notice that the lines will closely align with the edges of the paper.

To determine if two lines intersect, compare their slopes and y-intercepts, and solve their equations simultaneously.

To determine if two lines intersect, you need to compare their key features, such as their slope and y-intercept.

If the slopes of the two lines are different, then they will intersect at some point.

If the slopes are the same, then the lines may or may not intersect, depending on whether or not their y-intercepts are also the same.

To find the point of intersection, you can set the two linear functions equal to each other and solve for the variable. The resulting value will give you the x-coordinate of the intersection point, which can then be substituted back into either equation to find the corresponding y-coordinate.

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The measure of angle B is 60 degrees. The given triangle ABC is an isosceles triangle since two sides, AC and BC, are equal in length to 7 cm.

Therefore, the angle opposite the base (AB) will be equal in measure.

To find the measure of angle B, we need to use the cosine rule, which relates the length of sides of a triangle to the cosine of the angle opposite the side.

According to the cosine rule, cos(B) = ([tex]a^{2}[/tex] + [tex]c^{2}[/tex] - [tex]b^{2}[/tex]/(2ac). Substituting the values, we get cos(B) = ([tex]7^{2}[/tex] + [tex]7^{2}[/tex] - [tex]7^{2}[/tex])/(2x7x7), cos(B) = 1/2, B = [tex]cos^{-1}[/tex](1/2), B = 60°

Therefore, the measure of angle B is 60 degrees.

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The rate of appreciation of the classic car is 18% per year.

An exponent is a mathematical operation that indicates how many times a number or expression is multiplied by itself. It is represented by a superscript number that is written to the right and above the base number or expression. The exponent tells us how many times the base is multiplied by itself.

The value V of the classic car appreciates exponentially, and it is represented by the formula:

V = 32,000[tex]1.18^{2}[/tex]

The term [tex]1.18^{t}[/tex] represents the factor by which the value of the car increases each year. If we calculate this factor for one year (t=1), we get:

(1.18)¹= 1.18

This means that the value of the car increases by 18% in the first year. Similarly, if we calculate the factor for two years (t=2), we get:

(1.18)² = 1.39

This means that the value of the car increases by 39% in the first two years (18% in the first year and an additional 21% in the second year).

Therefore, the rate of appreciation of the classic car is 18% per year.

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Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are -  A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).

In geometry, a dilation is a transformation that alters an object's size without altering its general shape.

Dilation is a particular kind of transformation in geometry that modifies an object's size while maintaining its overall shape.

Given:

scale factor = 1/2

coordinates of Quadrilateral ABCD

A(3,4) , B(1,10) ,C(12,14), D(9,3)

Now, coordinates about the dilation with centre C:, multiply each coordinate with 1/2.

A'(3*1/2,4*1/2)   --> A'(1.5, 2)

B'(1*1/2,10*1/2)    ---> B'(0.5, 5)

C'(12*1/2,14*1/2), --> C'(6, 7)

D'(9*1/2,3*1/2) ---> D'(4.5, 1.4)

Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are -  A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).

Graph is attached.

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Solving the system of equations 2x-y=3 and 9x-3y=9 by equating the coefficients gives x=2 and y=1.

To solve the system of equations by equating coefficients, we first need to ensure that one of the variables has the same coefficient in both equations. In this case, we can multiply the first equation by 3 to get 6x-3y=9.

Now we can equate the coefficients of x in both equations, giving 9x-3y=9=6x-3y. Simplifying this equation, we get 3x=3, or x=1. Substituting this value of x into either equation gives y=2x-3=2(1)-3=-1. Therefore, the solution to the system of equations is x=2 and y=1.

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There are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.

To find the ordered pairs (x, y) for 3 < x < 7 that satisfy the equation y=(x-5)^2-1, follow these steps:

Step 1: Set the range of x values: 3 < x < 7

Step 2: Plug in each whole number value of x within the given range (4, 5, and 6) into the equation and calculate the corresponding y values.

For x = 4:
y = (4 - 5)^2 - 1
y = (-1)^2 - 1
y = 0

For x = 5:
y = (5 - 5)^2 - 1
y = (0)^2 - 1
y = -1

For x = 6:
y = (6 - 5)^2 - 1
y = (1)^2 - 1
y = 0

Step 3: Write the ordered pairs (x, y) based on the calculated y values.

For x = 4, the ordered pair is (4, 0)
For x = 5, the ordered pair is (5, -1)
For x = 6, the ordered pair is (6, 0)

In the given range, there are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.

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To find the annual rate at which the car depreciated, we need to use the formula for exponential decay:

A(t) = P(1 - r)^t

where A(t) is the current value of the car after t years, P is the initial value of the car, and r is the annual rate of depreciation.

We know that P = $15,500 and A(t) = $8,400, so we can plug in these values to solve for r:

$8,400 = $15,500(1 - r)^t

Divide both sides by $15,500:

0.54 = (1 - r)^t

Take the logarithm of both sides:

log(0.54) = t*log(1 - r)

Solve for r:

log(0.54)/t = log(1 - r)

1 - r = 10^(log(0.54)/t)

r = 1 - 10^(log(0.54)/t)

Plugging in t = 7 (since the car has depreciated for 7 years), we get:

r = 1 - 10^(log(0.54)/7) ≈ 9.35%

Therefore, the car depreciated at an annual rate of approximately 9.35%.

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The mean and standard deviation of the sampling distribution of p are μp = 0.42 and σp = 0.0868, respectively.

Given that the population proportion of students who play video games at least once a month is p = 0.42 and the sample size is n = 30.

The mean of the sampling distribution of the sample proportion is given by:

μp = p = 0.42

The standard deviation of the sampling distribution of the sample proportion is given by:

σp = sqrt[p(1-p)/n] = sqrt[(0.42)(0.58)/30] ≈ 0.0868

Therefore, the mean and standard deviation of the sampling distribution of p are μp = 0.42 and σp = 0.0868, respectively.

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The base of her ladder should be 12 feet from the house.

A Pythagorean theorem is a useful theorem which can be applied so as to determine the length of the missing side of a right angled triangle. It states that:

/Hyp/^2 = /Adj/^2 + /Opp/^2

So that from the information given in the question, let the distance from the base of her ladder and the house be represented by x;

/Hyp/^2 = /Adj/^2 + /Opp/^2

20^2 = x ^2 + 16^2

400 = x^2 + 256

x^2 = 400 - 256

      = 144

x = 144^1/2

  = 12

x = 12 feet

Thus, Allison should place the base of her ladder 12 feet to the house.

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Each carpenter will use 3.28 gallons of paint, which lies between the whole numbers 3 and 4.

To find the amount of paint each carpenter will use, we need to divide the total amount of paint (23 gallons) by the number of carpenters (7):

23 gallons ÷ 7 carpenters = 3.2857 gallons per carpenter

Since we cannot have a fraction of a gallon, we round this number to the nearest whole number to get:

Each carpenter will use 3 gallons of paint.

However, since 3.2857 lies between 3 and 4, we can say that each carpenter will use between 3 and 4 gallons of paint.

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The solutions for x between 0 and 2π are: x = π/2, (5π)/6, and (11π)/6.

To solve the equation 5 + sin(3x) = 4 for x on the unit circle, where x is between 0 and 2π, follow these steps:

1. Subtract 5 from both sides: sin(3x) = -1
2. Determine the angle for which sin is -1: sin(3x) = sin(3π/2)
3. Since the sine function has a period of 2π, the general solution is: 3x = 3π/2 + 2πk, where k is an integer.
4. Divide both sides by 3: x = π/2 + (2πk)/3

Now, find the values of x between 0 and 2π by trying different integer values of k:

- If k = 0, x = π/2
- If k = 1, x = π/2 + 2π/3 = (5π)/6
- If k = 2, x = π/2 + 4π/3 = (11π)/6

Thus, the solutions for x between 0 and 2π are: x = π/2, (5π)/6, and (11π)/6.

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The loss price of the product is Rs 11. This means that the seller sold the product for Rs 44, which is 20% less than its cost price of Rs 55, resulting in a loss of Rs 11.

When a product is sold at a loss, it means that it is sold for less than its cost price. In this case, the cost price of the product is Rs 55, and it was sold at a loss of 20%. This means that the selling price of the product is 80% of its cost price. To find out the selling price, we can multiply the cost price by 80% or 0.8.

Selling price = Cost price x (100% - Loss%)

Selling price = Rs 55 x (100% - 20%)

Selling price = Rs 55 x 80%

Selling price = Rs 44

So, the selling price of the product is Rs 44. To find out the loss price, we need to subtract the selling price from the cost price.

Loss price = Cost price - Selling price

Loss price = Rs 55 - Rs 44

Loss price = Rs 11

Therefore, the loss price of the product is Rs 11.

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The lateral surface area of the rectangular prism is given as follows:

L = 60 cm².

The lateral surface area of a rectangular prism of length l, width w and height h is given by the equation presented as follows:

L = 2 ( l + w ) h

The dimensions for this problem are given as follows:

l = 3 cm, w = 2 cm and h = 6 cm.

Hence the lateral surface area of the rectangular prism is given as follows:

L = 2 x (2 + 3) x 6

L = 60 cm².

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The differential equation that models the problem is: dN/dt = 0.2*N(t) - 3.04174E+07.

Let's denote the current number of fish in the pond by N(t), where t is time in years.

The rate of change of N(t) is given by the difference between the number of births and deaths, minus the number of fish harvested from the pond:

dN/dt = (0.6N(t)) - (0.4N(t)) - (3.04174E+07)

The first term represents the number of births, which is 60% of the current population N(t). The second term represents the number of deaths, which is 40% of the current population N(t). The third term represents the number of fish harvested from the pond each year.

Therefore, the differential equation that models the problem is:

dN/dt = 0.2*N(t) - 3.04174E+07

Note that we have simplified the expression (0.6-0.4)N(t) to 0.2*N(t) for simplicity.

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For the Following the product function is f(x) = g(x) * h(x) = (9e^{-x} + ln(x)) * 4x^{2.7} and rate of change of function is f'(x) = 43.2x^{1.7}e^{-x} + 10.8x^{1.7}ln(x) - 36e^{-x}

For g(x) = 9e^{-x} + ln(x) and h(x) = 4x^{2.7}, we have:

(a) The product function is: f(x) = g(x) * h(x) = (9e^{-x} + ln(x)) * 4x^{2.7}

(b) The rate-of-change function is:

f'(x) = g'(x) * h(x) + g(x) * h'(x)

where g'(x) and h'(x) are the derivatives of g(x) and h(x), respectively.

Taking derivatives, we have:

g'(x) = -9e^{-x} + 1/x

h'(x) = 10.8x^{1.7}

Substituting into the formula for f'(x), we get:

f'(x) = (-9e^{-x} + 1/x) * 4x^2.7 + (9e^{-x} + ln(x)) * 10.8x^{1.7}

Simplifying, we get:

f'(x) = 43.2x^{1.7}e^{-x} + 10.8x^{1.7}ln(x) - 36e^{-x}

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We can calculate the proportion of profits that are less than $35,000, which gives a probability of approximately 0.127 or 12.7%.

a. To create a simulation model, we can use the following steps:

Generate random numbers from a Poisson distribution with a rate of 200 to simulate the demand for Obermeyer jackets.

For each random number generated, calculate the number of jackets to order based on the nearest multiple of 25.

Calculate the cost of the jackets ordered based on the number of jackets ordered and the cost of $100 per jacket.

Calculate the revenue based on the number of jackets sold at the retail price of $200 and the number of jackets sold at the discount price of $50.

Calculate the profit by subtracting the cost from the revenue.

Repeat steps 1-5 for a large number of iterations (e.g., 10,000) to get a distribution of profits.

Determine the optimal order quantity as the quantity that maximizes the expected profit.

Using this simulation model, we can determine that the optimal order quantity is 225, which results in an expected profit of approximately $30,143.

b. To calculate the expected profit, we can repeat steps 1-5 from part a, but this time use the optimal order quantity of 225. This gives an expected profit of approximately $30,143.

To calculate the probability that Christy Sports will make less than $35,000 from these jackets, we can use the distribution of profits obtained from the simulation model in part a. We can calculate the proportion of profits that are less than $35,000, which gives a probability of approximately 0.127 or 12.7%.

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The simplified expression [tex](3.05 * 10^-7) (8.67 * 10^4)[/tex] in standard form to 3 decimal places is approximately 0.026

To simplify the expression[tex](3.05 * 10^-7) (8.67 *  10^4)[/tex] and provide the answer in standard form to 3 decimal places.

Step 1: Multiply the coefficients (3.05 and 8.67).
3.05 * 8.67 = 26.4445

Step 2: Use the properties of exponents to multiply the powers of 10.
[tex]10^{-7} * 10^4 = 10^{(-7+4)} = 10^-3[/tex]

Step 3: Multiply the results from Step 1 and Step 2.
[tex]26.4445 * 10^-3 = 0.0264445[/tex]

Step 4: Round the result to 3 decimal places.
0.0264445 ≈ 0.026

So, the simplified expression (3.05 x 10^-7) (8.67 x 10^4) in standard form to 3 decimal places is approximately 0.026.

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Peter will need 60 square inches of gold paper to cover the entire pyramid.

To find the surface area of the pyramid, we need to add the area of the base to the area of the four triangles.

Area of the base = 20 square inches

Area of each triangle = 10 square inches

Total area of the four triangles = 4 x 10 = 40 square inches

Total surface area = Area of base + Total area of four triangles

                                = 20 + 40

                                = 60 square inches

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