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GED Math : Volume of a Cylinder

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Example Questions

Example Question #1 : Volume Of A Cylinder

One cubic foot is equal to (about) 7.5 gallons.

A circular swimming pool has diameter 60 feet and depth five feet throughout. Using the above conversion factor, how many gallons of water does it hold?

Use 3.14 for $\pi$ $\displaystyle \pi$.

Possible Answers:

$24,320\textrm{ gal}$ $\displaystyle 24,320\textrm{ gal}$

$105,975\textrm{ gal }$ $\displaystyle 105,975\textrm{ gal }$

$211,950\textrm{ gal }$ $\displaystyle 211,950\textrm{ gal }$

$7,065 \textrm{ gal }$ $\displaystyle 7,065 \textrm{ gal }$

Correct answer:

$105,975\textrm{ gal }$ $\displaystyle 105,975\textrm{ gal }$

Explanation:

The pool can be seen as a cylinder with depth (or height) 5 feet, and a base with diameter 60 feet - and radius half this, or 30 feet. The capacity of the pool is the volume of this cylinder, which is

$V = \pi r ^{2} h = 3.14 \times 30^{2} \times 5 = 14,130$ $\displaystyle V = \pi r ^{2} h = 3.14 \times 30^{2} \times 5 = 14,130$ cubic feet.

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

$14,130 \times 7.5 = 105,975$ $\displaystyle 14,130 \times 7.5 = 105,975$ gallons

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Example Question #2 : Volume Of A Cylinder

A cylindrical bucket is one foot high and one foot in diameter. It is filled with water, which is then emptied into an empty barrel three feet high and two feet in diameter. What percent of the barrel has been filled?

Possible Answers:

$12 \%$ $\displaystyle 12 \%$

$8 \frac{1}{3} \%$ $\displaystyle 8 \frac{1}{3} \%$

$18 \%$ $\displaystyle 18 \%$

$16\frac{2}{3} \%$ $\displaystyle 16\frac{2}{3} \%$

Correct answer:

$8 \frac{1}{3} \%$ $\displaystyle 8 \frac{1}{3} \%$

Explanation:

The volume of a cylinder is

$V = \pi r^{2}h$ $\displaystyle V = \pi r^{2}h$

The bucket has height h = 1 $\displaystyle h = 1$ and diameter 1, and,subsequently, radius $r = \frac{1}{2}$ $\displaystyle r = \frac{1}{2}$; its volume is

$V = \pi r^{2}h = \pi \cdot \left (\frac{1}{2} \right ) ^{2} \cdot 1 =\frac{1}{4} \pi$ $\displaystyle V = \pi r^{2}h = \pi \cdot \left (\frac{1}{2} \right ) ^{2} \cdot 1 =\frac{1}{4} \pi$ cubic feet

The barrel has height h = 3 $\displaystyle h = 3$ and diameter 2,and, subsequently, radius r = 1 $\displaystyle r = 1$; its volume is

$V = \pi r^{2}h = \pi \cdot 1 ^{2} \cdot 3= 3\pi$ $\displaystyle V = \pi r^{2}h = \pi \cdot 1 ^{2} \cdot 3= 3\pi$

The volume of the bucket is

$\frac{\frac{1}{4}\pi}{3 \pi} \times 100 \%$ $\displaystyle \frac{\frac{1}{4}\pi}{3 \pi} \times 100 \%$

$=\left ( \frac{1}{4}\pi \div 3 \pi \right )\times 100 \%$ $\displaystyle =\left ( \frac{1}{4}\pi \div 3 \pi \right )\times 100 \%$

$=\left ( \frac{\pi}{4}\times \frac{1}{3 \pi} \right )\times 100\%$ $\displaystyle =\left ( \frac{\pi}{4}\times \frac{1}{3 \pi} \right )\times 100\%$

$= \frac{1}{12} \times 100\% = 8 \frac{1}{3} \%$ $\displaystyle = \frac{1}{12} \times 100\% = 8 \frac{1}{3} \%$

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Example Question #1 : Volume Of A Cylinder

A water tank takes the shape of a closed cylinder whose exterior has height 30 feet and a base with radius 10 feet; the tank is three inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

You may use 3.14 for $\pi$ $\displaystyle \pi$.

Possible Answers:

$9,400 \textrm{ ft}^{3}$ $\displaystyle 9,400 \textrm{ ft}^{3}$

$9,000 \textrm{ ft}^{3}$ $\displaystyle 9,000 \textrm{ ft}^{3}$

$8,800 \textrm{ ft}^{3}$ $\displaystyle 8,800 \textrm{ ft}^{3}$

$8,400 \textrm{ ft}^{3}$ $\displaystyle 8,400 \textrm{ ft}^{3}$

Correct answer:

$8,800 \textrm{ ft}^{3}$ $\displaystyle 8,800 \textrm{ ft}^{3}$

Explanation:

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

$30 - 2 \times 0.25 = 30 - 0.5 = 29.5$ $\displaystyle 30 - 2 \times 0.25 = 30 - 0.5 = 29.5$ inches.

The radius of the interior of the tank is

$\displaystyle 10 - 0.25 = 9.75$ inches.

The amount of water the tank holds is the volume of the interior of the tank, which is

$V = \pi r^{2}h$ $\displaystyle V = \pi r^{2}h$

$V \approx 3.14 \cdot 9.75 ^{2} \cdot 29.5$ $\displaystyle V \approx 3.14 \cdot 9.75 ^{2} \cdot 29.5$

$V \approx 3.14 \cdot 9.75 ^{2} \cdot 29.5 \approx 8,806$ $\displaystyle V \approx 3.14 \cdot 9.75 ^{2} \cdot 29.5 \approx 8,806$ cubic feet.

This rounds to 8,800 cubic feet.

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Example Question #3 : Volume Of A Cylinder

Barrel

The above diagram is one of a cylindrical tub. The company wants to make a cylindrical tub with three times the volume, but whose base is only twice the radius. How high should this new tub be?

Possible Answers:

$26 \frac{2}{3}\textrm{ in}$ $\displaystyle 26 \frac{2}{3}\textrm{ in}$

$53 \frac{1}{3}\textrm{ in}$ $\displaystyle 53 \frac{1}{3}\textrm{ in}$

$60\textrm{ in}$ $\displaystyle 60\textrm{ in}$

$30\textrm{ in}$ $\displaystyle 30\textrm{ in}$

Correct answer:

$30\textrm{ in}$ $\displaystyle 30\textrm{ in}$

Explanation:

The volume of the given tub can be expressed using the following formula, setting r = 25 $\displaystyle r = 25$ and h= 40 $\displaystyle h= 40$:

$V = \pi r^{2} h$ $\displaystyle V = \pi r^{2} h$

$V = \pi \cdot 25^{2} \cdot 40$ $\displaystyle V = \pi \cdot 25^{2} \cdot 40$

$V = 25,000 \pi$ $\displaystyle V = 25,000 \pi$ cubic inches.

The new tub should have three times this volume, or

$V = 25,000 \pi \cdot 3 = 75,000 \pi$ $\displaystyle V = 25,000 \pi \cdot 3 = 75,000 \pi$ cubic inches.

The radius is to be twice that of the above tub, which will be

$r = 25 \cdot 2 = 50$ $\displaystyle r = 25 \cdot 2 = 50$ inches.

The height can therefore be calculated as follows:

$\pi r^{2} h = V$ $\displaystyle \pi r^{2} h = V$

$\pi \cdot 50^{2} \cdot h = 75,000 \pi$ $\displaystyle \pi \cdot 50^{2} \cdot h = 75,000 \pi$

$2,500 \pi h = 75,000 \pi$ $\displaystyle 2,500 \pi h = 75,000 \pi$

$h = \frac{75,000 \pi}{2,500 \pi} = 30$ $\displaystyle h = \frac{75,000 \pi}{2,500 \pi} = 30$ inches

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Example Question #1 : Volume Of A Cylinder

Barrel

Refer to the cylinder in the above diagram.

A cone has twice the volume and twice the height of the cylinder. What is the radius of the base of the cone (nearest tenth of an inch, if applicable)?

Possible Answers:

$43.3 \textup{ in}$ $\displaystyle 43.3 \textup{ in}$

$50\textup{ in}$ $\displaystyle 50\textup{ in}$

$75 \textup{ in}$ $\displaystyle 75 \textup{ in}$

$35.4\textup{ in}$ $\displaystyle 35.4\textup{ in}$

Correct answer:

$43.3 \textup{ in}$ $\displaystyle 43.3 \textup{ in}$

Explanation:

The formula for the volume of the cylinder is

$v = \pi r^{2} h$ $\displaystyle v = \pi r^{2} h$

where $\displaystyle r$ and $\displaystyle h$ are the base radius and height of the cylinder.

Set $\displaystyle r = 25, h = 40$ in the formula to find the volume of the cylinder:

$v = \pi r^{2} h$ $\displaystyle v = \pi r^{2} h$

$v = \pi \cdot 25^{2} \cdot 40 = \pi \cdot 625 \cdot 40 = 25,000 \pi$ $\displaystyle v = \pi \cdot 25^{2} \cdot 40 = \pi \cdot 625 \cdot 40 = 25,000 \pi$

The cylinder will have volume twice this, or $50,000 \pi$ $\displaystyle 50,000 \pi$, and height twice the height of the cylinder, which is 80 inches.

The formula for the volume of the cone is

$V =\frac{ \pi r^{2} h}{3}$ $\displaystyle V =\frac{ \pi r^{2} h}{3}$,

so we set $V = 50,000 \pi$ $\displaystyle V = 50,000 \pi$ and h = 80 $\displaystyle h = 80$ and solve for $\displaystyle r$:

$V =\frac{ \pi r^{2} h}{3}$ $\displaystyle V =\frac{ \pi r^{2} h}{3}$

$\frac{ \pi r^{2} \cdot 80}{3} = 50,000 \pi$ $\displaystyle \frac{ \pi r^{2} \cdot 80}{3} = 50,000 \pi$

$\frac{ \pi r^{2} \cdot 80}{3} \cdot \frac{3}{80 \pi} = 50,000 \pi \cdot \frac{3}{80 \pi}$ $\displaystyle \frac{ \pi r^{2} \cdot 80}{3} \cdot \frac{3}{80 \pi} = 50,000 \pi \cdot \frac{3}{80 \pi}$

$r^{2} = \frac{150,000 \pi}{80 \pi} = 1,875$ $\displaystyle r^{2} = \frac{150,000 \pi}{80 \pi} = 1,875$

$r = \sqrt{1,875} \approx 43.3$ $\displaystyle r = \sqrt{1,875} \approx 43.3$ inches.

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Example Question #1 : Volume Of A Cylinder

Barrel

The above diagram is one of a cylindrical tub. The tub is holding water at 40% capacity. To the nearest cubic foot, how much more water can it hold?

Possible Answers:

$75 \textup{ ft}^{3}$ $\displaystyle 75 \textup{ ft}^{3}$

$18 \textup{ ft}^{3}$ $\displaystyle 18 \textup{ ft}^{3}$

$45 \textup{ ft}^{3}$ $\displaystyle 45 \textup{ ft}^{3}$

$27 \textup{ ft}^{3}$ $\displaystyle 27 \textup{ ft}^{3}$

Correct answer:

$27 \textup{ ft}^{3}$ $\displaystyle 27 \textup{ ft}^{3}$

Explanation:

The volume of the cylinder can be calculated using the following formula, setting r = 25 $\displaystyle r = 25$ and h= 40 $\displaystyle h= 40$:

$V = \pi r^{2} h$ $\displaystyle V = \pi r^{2} h$

$V \approx 3.14 \cdot 25^{2} \cdot 40 \approx 78,539.8$ $\displaystyle V \approx 3.14 \cdot 25^{2} \cdot 40 \approx 78,539.8$ cubic inches.

The tub is 40% full, so it is 60% empty; it can hold

$78,539.8 \times 0.60 \approx 47,123.9$ $\displaystyle 78,539.8 \times 0.60 \approx 47,123.9$ more cubic inches of water.

The problem asks for the number of cubic feet, so divide by the cube of 12, or 1,728:

$47,123.9 \div 1,728 \approx 27.2$ $\displaystyle 47,123.9 \div 1,728 \approx 27.2$

The correct response is 27 cubic feet.

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Example Question #2 : Volume Of A Cylinder

A circular swimming pool has diameter 40 feet and depth six feet throughout. How many cubic feet of water does it hold? (nearest cubic foot)

Use 3.14 as the value of $\pi$ $\displaystyle \pi$.

Possible Answers:

$1,507 \textrm{ ft}^{3}$ $\displaystyle 1,507 \textrm{ ft}^{3}$

$7,536 \textrm{ ft}^{3}$ $\displaystyle 7,536 \textrm{ ft}^{3}$

$754 \textrm{ ft}^{3}$ $\displaystyle 754 \textrm{ ft}^{3}$

$15,072 \textrm{ ft}^{3}$ $\displaystyle 15,072 \textrm{ ft}^{3}$

Correct answer:

$7,536 \textrm{ ft}^{3}$ $\displaystyle 7,536 \textrm{ ft}^{3}$

Explanation:

The pool can be seen as a cylinder with depth (or height) six feet and a base with diameter 40 feet - and radius half this, or 20 feet. The capacity of the pool is the volume of this cylinder, which is

$V = \pi r ^{2} h \approx 3.14 \times 20^{2} \times 6 = 7,536$ $\displaystyle V = \pi r ^{2} h \approx 3.14 \times 20^{2} \times 6 = 7,536$ cubic feet.

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Example Question #1 : Volume Of A Cylinder

A circular swimming pool has diameter 20 meters and depth 2.5 meters throughout. How many cubic meters of water does it hold?

Use 3.14 for $\pi$ $\displaystyle \pi$.

Possible Answers:

$785 \textrm{ m}^{3}$ $\displaystyle 785 \textrm{ m}^{3}$

$314 \textrm{ m}^{3}$ $\displaystyle 314 \textrm{ m}^{3}$

$1,570 \textrm{ m}^{3}$ $\displaystyle 1,570 \textrm{ m}^{3}$

$157 \textrm{ m}^{3}$ $\displaystyle 157 \textrm{ m}^{3}$

Correct answer:

$785 \textrm{ m}^{3}$ $\displaystyle 785 \textrm{ m}^{3}$

Explanation:

The pool can be seen as a cylinder with depth (or height) 2.5 m, a base with diameter 20 m, and a radius of half this, or 10 m. The capacity of the pool is the volume of this cylinder, which is

$V = \pi r ^{2} h = 3.14 \times 10^{2} \times 2.5 = 785$ $\displaystyle V = \pi r ^{2} h = 3.14 \times 10^{2} \times 2.5 = 785$ cubic meters.

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Example Question #3 : Volume Of A Cylinder

Find the volume of a cylinder with a radius of 2, and a height of 11.

Possible Answers:

$52\pi$ $\displaystyle 52\pi$

$22\pi$ $\displaystyle 22\pi$

$60\pi$ $\displaystyle 60\pi$

$44\pi$ $\displaystyle 44\pi$

$33\pi$ $\displaystyle 33\pi$

Correct answer:

$44\pi$ $\displaystyle 44\pi$

Explanation:

Write the volume for the cylinder.

$V=\pi r^2 h$ $\displaystyle V=\pi r^2 h$

Substitute the dimensions.

$V=\pi r^2 h = \pi (2)^2(11) = \pi (4)(11) = 44\pi$ $\displaystyle V=\pi r^2 h = \pi (2)^2(11) = \pi (4)(11) = 44\pi$

The answer is: $44\pi$ $\displaystyle 44\pi$

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Example Question #2 : Volume Of A Cylinder

Find the volume of a cylinder with a radius of 2, and a height of 15.

Possible Answers:

$120\pi$ $\displaystyle 120\pi$

$450\pi$ $\displaystyle 450\pi$

$90\pi$ $\displaystyle 90\pi$

$30\pi$ $\displaystyle 30\pi$

$60\pi$ $\displaystyle 60\pi$

Correct answer:

$60\pi$ $\displaystyle 60\pi$

Explanation:

write the formula for the volume of a cylinder.

$V=\pi r^2 h$ $\displaystyle V=\pi r^2 h$

Substitute the radius and height.

$V=\pi (2)^2 (15) = 60\pi$ $\displaystyle V=\pi (2)^2 (15) = 60\pi$

The answer is: $60\pi$ $\displaystyle 60\pi$

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GED Math : Volume of a Cylinder

Study concepts, example questions & explanations for GED Math

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Example Question #1 : Volume Of A Cylinder

Example Question #2 : Volume Of A Cylinder

Example Question #1 : Volume Of A Cylinder

Example Question #3 : Volume Of A Cylinder

Example Question #1 : Volume Of A Cylinder

Example Question #1 : Volume Of A Cylinder

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