Volume of a Cylinder - GED Math

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Question

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 $\pi$ .

Answer

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

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

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Question

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 $\pi$ .

Answer

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

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

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

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

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Question

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?

Answer

The volume of a cylinder is

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

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

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

The barrel has height and diameter 2,and, subsequently, radius ; its volume is

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

The volume of the bucket is

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

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

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

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

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Question

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 $\pi$ .

Answer

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

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

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Question

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 $\pi$ .

Answer

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

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

The radius of the interior of the tank is

inches.

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

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

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

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

This rounds to 8,800 cubic feet.

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Question

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?

Answer

The volume of the cylinder can be calculated using the following formula, setting and :

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

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

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

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

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

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

The correct response is 27 cubic feet.

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Question

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?

Answer

The volume of the given tub can be expressed using the following formula, setting and :

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

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

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

The new tub should have three times this volume, or

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

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

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

The height can therefore be calculated as follows:

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

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

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

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

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Question

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)?

Answer

The formula for the volume of the cylinder is

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

where and are the base radius and height of the cylinder.

Set in the formula to find the volume of the cylinder:

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

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

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

The formula for the volume of the cone is

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

so we set $V = 50,000 \pi$ and and solve for :

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

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

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

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

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

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Question

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

Answer

Write the volume for the cylinder.

$V=\pi r^2 h$

Substitute the dimensions.

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

The answer is: $44\pi$

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Question

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

Answer

write the formula for the volume of a cylinder.

$V=\pi r^2 h$

Substitute the radius and height.

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

The answer is: $60\pi$

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Question

Find the volume of a cylinder with a base area of 6 and a height of 4.

Answer

The base of the cylinder is a circle, and the area of the base is already given.

Write the volume of the cylinder.

$V=\pi r^2 h = B_{area}h$

Substitute the known values to determine the volume.

The volume is:

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Question

If the circular base area of a cylinder is , what is the volume of the cylinder if the height is ?

Answer

Write the formula for the volume of the cylinder.

$V=\pi r^2 h$

The area of the circle is $A =\pi r^2$ .

We can replace this term with .

Substitute the known terms.

$V=(\pi r^2 )h = (10)(5) = 50$

The volume is:

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Question

Find the volume of a cylinder with the following measurements:

Diameter: 8in

Height: 12in

Answer

To find the volume of a cylinder, we will use the following formula:

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

where r is the radius, and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 8in. We also know the diameter is two times the radius. Therefore, the radius is 4in.

We know the height of the cylinder is 12in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (4\text{in})^2 \cdot 12\text{in}$

$V = \pi \cdot 16\text{in}^2 \cdot 12\text{in}$

$V = \pi \cdot 192\text{in}^3$

$V = 192\pi \text{ in}^3$

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Question

Find the volume of a cylinder with a height of 14in and a diameter of 8in.

Answer

To find the volume of a cylinder, we will use the formula

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

where r is the radius, and h is the height of the cylinder.

Now, we know the height of the cylinder is 14in.

We know the diameter of the cylinder is 8in. We also know the diameter is two times the radius. Therefore, the radius is 4in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (4\text{in})^2 \cdot 14\text{in}$

$V = \pi \cdot 16\text{in}^2 \cdot 14\text{in}$

$V = \pi \cdot 224\text{in}^3$

$V = 224\pi \text{ in}^3$

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Question

What is the volume of a cylinder if the base diameter is 10, and the height is 5?

Answer

Write the formula for the volume of a cylinder.

$V=\pi r^2h$

The radius is half the diameter.

Substitute the terms in the equation.

$V=\pi5^2(5) = 125 \pi$

The answer is: $125 \pi$

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Question

What is the volume of a cylinder with a base circumference of $2\pi$ and a height of ?

Answer

To determine the radius of the base, write the circumference formula. The base represents a circle.

$C=2\pi r$

Substitute the circumference.

$2\pi =2\pi r$

Divide by $2\pi$ on both sides.

$\frac{2\pi}{2\pi} =\frac{2\pi r}{2\pi}$

Write the formula for the volume of a cylinder.

$V=\pi r^2h$

Substitute the radius and height.

$V=\pi (1)^2(4) = 4\pi$

The answer is: $4\pi$

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Question

Find the volume of a cylinder with the following measurements:

Diameter: 10in

Height: 14in

Answer

To find the volume of a cylinder, we will use the following formula:

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

where r is the radius, and h is the height of the cylinder.

Now, we know the diameter of the cylinder is 10in. We also know the diameter is two times the radius. Therefore, the radius is 5in.

We know the height of the cylinder is 14in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (5\text{in})^2 \cdot 14\text{in}$

$V = \pi \cdot 25\text{in}^2 \cdot 14\text{in}$

$V = \pi \cdot 350\text{in}^3$

$V = 350\pi \text{ in}^3$

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Question

Find the volume of a cylinder with a height of 11in and a diameter of 6in.

Answer

To find the volume of a cylinder, we will use the formula

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

where r is the radius, and h is the height of the cylinder.

Now, we know the height of the cylinder is 11in.

We know the diameter of the cylinder is 6in. We also know the diameter is two times the radius. Therefore, the radius is 3in.

Knowing all of this, we can substitute into the formula. We get

$V = \pi \cdot (3\text{in})^2 \cdot 11\text{in}$

$V = \pi \cdot 9\text{in}^2 \cdot 11\text{in}$

$V = \pi \cdot 99\text{in}^3$

$V = 99\pi \text{ in}^3$

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Question

Determine the volume of a cylinder with a radius of 5, and a height of 20.

Answer

Write the formula for the volume of a cylinder.

$V=\pi r^2h$

Substitute the radius and the height into the formula.

$V=\pi (5^2)(20) = \pi (25)(20) = 500 \pi$

The answer is: $500 \pi$

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Question

Find the volume of a cylinder with a radius of 8, and a height of 20.

Answer

Write the formula for the volume of a cylinder.

$V=\pi r^2h$

Substitute the radius and height into the equation.

$V=\pi( 8)^2(20)= \pi( 64)(20)$

The answer is: $1280\pi$

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