Volume - SAT Subject Test in Math II

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Question

One cubic meter is equal to one thousand liters.

A circular swimming pool is meters in diameter and meters deep throughout. How many liters of water does it hold?

Answer

The pool can be seen as a cylinder with depth (or height) , and a base with diameter - and, subsequently, radius half this, or $\frac{d}{2}$ . The volume of the pool in cubic meters is

$V = \pi r^{2}h = \pi \left ( \frac{d}{2} \right )^{2} \cdot f = \pi \cdot \frac{d^{2}}{4} \cdot f = \frac{ \pi d^{2}f}{4}$

Multiply this number of cubic meters by 1,000 liters per cubic meter:

$V = \frac{ \pi d^{2}f}{4} \cdot \1,000 = 250\pi d^{2}f$

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Question

A circular swimming pool has diameter 80 feet and depth five feet throughout. To the nearest thousand, how many gallons of water does it hold?

Use the conversion factor: One cubic foot = 7.5 gallons.

Answer

The pool can be seen as a cylinder with depth (or height) 5 feet, and a base with diameter 80 feet - and radius half this, or 40 feet. The capacity of the pool is the volume of this cylinder, which is

$V = \pi r ^{2} h \approx 3.14 \times 40^{2} \times 5 \approx 25,120$ cubic feet.

One cubic foot is equal to 7.5 gallons, so multiply:

$25,120 \times 7.5 = 188,400$ gallons

This rounds to 188,000 gallons.

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Question

The above depicts a rectangular swimming pool for an apartment.

On the left and right edges, the pool is three feet deep; the dashed line at the very center represents the line along which it is eight feet deep. Going from the left to the center, its depth increases uniformly; going from the center to the right, its depth decreases uniformly.

In cubic feet, how much water does the pool hold?

Answer

The pool can be looked at as a pentagonal prism with "height" 35 feet and its bases the following shape (depth exaggerated):

This is a composite of two trapezoids, each with bases 3 feet and 8 feet and height 25 feet; the area of each is

$A = \frac{1}{2} (3+8) \cdot 25 = 137.5$ square feet.

The area of the base is twice this, or

$B = 137.5 \cdot 2 = 275$ square feet.

The volume of a prism is its height times the area of its base, or

$V = 275 \cdot 35 = 9,625$ cubic feet, the capacity of the pool.

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Question

A water tank takes the shape of a sphere whose exterior has radius 16 feet; the tank is three inches thick throughout. To the nearest hundred, give the surface area of the interior of the tank in square feet.

Answer

Three inches is equal to 0.25 feet, so the radius of the interior of the tank is

feet.

The surface area of the interior of the tank can be calculated using the formula

$SA = 4 \pi r^{2}$

$SA \approx 4 \cdot 3.14 \cdot 15.75 ^{2} \approx 3,116$ ,

which rounds to 3,100 square feet.

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Question

A water tank takes the shape of a closed rectangular prism whose exterior has height 30 feet, length 20 feet, and width 15 feet. Its walls are one foot thick throughout. How many cubic feet of water does the tank hold?

Answer

The height, length, and width of the interior tank are each two feet less than the corresponding dimension of the exterior of the tank, so the dimensions of the interior are 28, 18, and 13 feet. Multiply these to get the volume:

$V = 28 \cdot 18 \cdot 13 = 6,552$ cubic feet.

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Question

A circular swimming pool has diameter 40 meters and depth meters throughout. Which of the following expressions gives the amount of water it holds, in cubic meters?

Answer

The pool can be seen as a cylinder with depth (or height) , and a base with diameter 40 m - and radius half this, or . The capacity of the pool is the volume of this cylinder, which is

$V = \pi r^{2}h = \pi \cdot 20^{2}\cdot t = 400 \pi t$

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Question

One cubic meter is equal to one thousand liters.

A rectangular swimming pool is meters deep throughout and meters wide. Its length is ten meters greater than twice its width. How many liters of water does the pool hold?

Answer

Since the length of the pool is ten meters longer than twice its width , its length is .

The inside of the pool can be seen as a rectangular prism, and as such, its volume in cubic feet can be calculated as the product of its length, width, and height (or depth). This product is

$V = D \cdot (2W + 10) \cdot W$

$V = 2DW^{2} + 10DW$

Multiply this by the conversion factor 1,000, and its volume in liters is

$V =1,000 \left ( 2DW^{2} + 10DW \right )$

$V = 2,000 DW^{2} + 10,000 DW$

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Question

The bottom surface of a rectangular prism has area 100; the right surface has area 200; the rear surface has area 300. Give the volume of the prism (nearest whole unit), if applicable.

Answer

Let the dimensions of the prism be , , and .

Then, , , and .

From the first and last equations, dividing both sides, we get

$\frac{WH}{LW} = \frac{300}{100}$

$\frac{H}{L} = 3$

Along with the second equation, multiply both sides:

$\frac{H}{L} \cdot LH = 3 \cdot 200$

$H ^{2}= 600$

Taking the square root of both sides and simplifying, we get

$H = \sqrt{600} = \sqrt{100} \cdot \sqrt{6} = 10 \sqrt{6}$

Now, substituting and solving for the other two dimensions:

$L \cdot 10 \sqrt{6} = 200$

$L = \frac{200}{10 \sqrt{6}} = \frac{20}{\sqrt{6}}$

$W \cdot 10 \sqrt{6} = 300$

$W = \frac{300}{10 \sqrt{6}} = \frac{30 }{\sqrt{6}}$

Now, multiply the three dimensions to obtain the volume:

$= \frac{20}{\sqrt{6}} \cdot 10 \sqrt{6} \cdot \frac{30 }{\sqrt{6}}$

$= \frac{20}{\sqrt{6}} \cdot\frac{ 10 \sqrt{6}}{1} \cdot \frac{30 }{\sqrt{6}}$

$= \frac{ 6,000 \sqrt{6}} {6}$

$= 1,000 \sqrt{6}$

$\approx 1,000 \cdot 2.449$

$\approx 2,449$

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Question

The width of a box is two-thirds its height and three-fifths its length. The volume of the box is 6 cubic meters. To the nearest centimeter, give the width of the box.

Answer

Call , , and the length, width, and height of the crate.

The width is two-thirds the height, so

$W = \frac{2}{3} H$ .

Equivalently,

$\frac{3} {2} \cdot \frac{2}{3} H = \frac{3} {2} W$

$H = \frac{3} {2} W$

The width is three-fifths the length, so

$W = \frac{3}{5} L$ .

Equivalently,

$\frac{5} {3} \cdot \frac{3}{5} L = \frac{5} {3} \cdot W$

$L = \frac{5} {3 } W$

The dimensions of the crate in terms of are , $\frac{3} {2} W$ , and $\frac{5} {3 } W$ . The volume is their product:

,

Substitute:

$\frac{5} {3 } W \cdot W \cdot \frac{3} {2} W = V$

$\frac{5} {3 } \cdot \frac{3} {2} \cdot W \cdot W \cdot W = 6$

$\frac{5} {2} \cdot W ^{3}= 6$

$\frac{2} {5} \cdot \frac{5} {2} \cdot W ^{3}= \frac{2} {5} \cdot 6$

$W ^{3}= \frac{12} {5}$

Taking the cube root of both sides:

$W =\sqrt[3]{ \frac{12} {5}}$

$W \approx 1.339$ meters.

Since one meter comprises 100 centimeters, multiply by 100 to convert to centimeters:

$1.339 \cdot 100 = 133.9$ centimeters,

which rounds to 134 centimeters.

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Question

The shaded face of the rectangular prism in the above diagram is a square. The volume of the prism is ; give the value of in terms of .

Answer

The volume of a rectangular prism is the product of its length, its width, and its height; that is,

Since the shaded face of the prism is a square, we can set , and ; substituting and solving for :

$x \cdot 25 \cdot x = V$

$25x^{2} = V$

$\frac{25x^{2}}{25} = \frac{V}{25}$

$x^{2} = \frac{V}{25}$

Taking the positive square root of both sides, and simplifying the expression on the right using the Quotient of Radicals Rule:

$x =\sqrt{ \frac{V}{25}} = \frac{\sqrt{V}}{\sqrt{25}} = \frac{\sqrt{V}}{5}$

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Question

Find the volume of a sphere with a diameter of 10.

Answer

The surface area of a sphere is found using the formula $A=\frac{4}{3}\pi r^3$ . We are given the diameter of the circle and so we have to use it to find the radius (r).

$Radius=\frac{Diameter}{2}=\frac{10}{2}=5$

Plug r into the formula to find the surface area

$A=\frac{4}{3}\pi 5^3=\frac{4}{3}\pi 125=166.67\pi$

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Question

Determine the volume of the cube with a side length of .

Answer

Write the formula for the volume of a cube.

Substitute the length into the formula.

The volume is:

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Question

Billy has a ice cream cone that consists of a cone and hemisphere. Suppose the cone has a height of 4 inches, and the radius of the hemisphere is 2 inches. Assuming that the combined shape is not irregular, what is the total volume?

Answer

Write the volume for a cone.

$V_{cone } = \frac{1}{3}\pi r^2 h$

Substitute the radius and height. The radius is 2.

$V_{cone } = \frac{1}{3}\pi (2)^2 (4) = \frac{16}{3}\pi$

Write the volume for a hemisphere. This should be half the volume of the full sphere.

$V_{hemisphere} = \frac{1}{2}[ \frac{4}{3}\pi r^3] = \frac{2}{3}\pi r^3$

Substitute the radius.

$V_{hemisphere}= \frac{2}{3}\pi (2)^3 = \frac{16}{3}\pi$

Add the volumes of the cone and hemisphere to determine the total volume.

$\frac{16}{3}\pi+\frac{16}{3}\pi$

The answer is: $\frac{32}{3}\pi$

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Question

Find the volume of a sphere with a diameter of $\frac{\sqrt[3]{2}}{2}$ .

Answer

Divide the diameter by two to get the radius. This is also the same as multiplying the diameter by one-half.

$r =\frac{\sqrt[3]{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt[3]{2}}{4}$

Write the formula for the volume of the sphere.

$V=\frac{4}{3}\pi r^3$

Substitute the radius.

$V=\frac{4}{3}\pi ( \frac{\sqrt[3]{2}}{4})^3$

Simplify the terms.

$\frac{4}{3}\pi ( \frac{\sqrt[3]{2}}{4})^3 = \frac{4}{3}\pi ( \frac{\sqrt[3]{2}}{4})( \frac{\sqrt[3]{2}}{4})( \frac{\sqrt[3]{2}}{4}) = \frac{2}{3(4)(4)}\pi$

The answer is: $\frac{1}{24}\pi$

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Question

If the side of a cube is , what must be the volume?

Answer

Write the formula for the volume of a cube.

Substitute the side length. When we are multiplying common bases with exponents, we are adding the exponents instead.

The answer is:

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Question

Determine the volume of a cube if the side length is $2\sqrt6$ .

Answer

Write the formula for the volume of a cube.

Substitute the side length into the equation.

$V= (2\sqrt6)^3 = (2\sqrt6)(2\sqrt6)(2\sqrt6) = 8\cdot 6\cdot \sqrt6$

The answer is: $48\sqrt6$

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Question

The radius and the height of a cylinder are equal. If the volume of the cylinder is , what is the diameter of the cylinder?

Answer

Recall how to find the volume of a cylinder:

$\text{Volume}=\pi r^2 h$

Since we know that the radius and the height are equal, we can rewrite the equation:

$\text{Volume}=\pi r^2 (r)=\pi r^3$

Using the given volume, find the length of the radius.

$13=\pi r^3$

Since the question asks you to find the diameter, multiply the radius by two.

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Question

Determine the volume of the cube if the side lengths are $\frac{2}{5}$ .

Answer

The volume of a cube is:

Substitute the dimensions.

$V =(\frac{2}{5})(\frac{2}{5})(\frac{2}{5}) = \frac{8}{125}$

The answer is: $\frac{8}{125}$

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Question

Determine the volume of a sphere with a diameter of 6.

Answer

The radius is half the diameter, which is three.

Write the formula for the volume of a sphere.

$V=\frac{4}{3}\pi r^3$

Substitute the radius.

$V=\frac{4}{3}\pi (3)^3 = 36\pi$

The answer is: $36\pi$

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Question

Find the volume of a sphere with a radius of $2\pi^4$ .

Answer

Write the formula for the sphere.

$V=\frac{4}{3} \pi r^3$

Substitute the radius.

$V=\frac{4}{3} \pi (2\pi^4)^3 = \frac{4}{3} \pi (2\pi^4)(2\pi^4)(2\pi^4)$

The answer is: $\frac{32}{3}\pi ^{13}$

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