Volume of a Sphere - GED Math

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Question

A water tank takes the shape of a sphere whose exterior has radius 18 feet; the tank is three inches thick throughout. To the nearest hundred, how many cubic feet of water does the tank hold?

Use 3.14 for $\pi$ .

Answer

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

feet.

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

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

$V \approx \frac{4}{3} \cdot 3.14 \cdot 17.75^{3} \approx 23,413$ ,

which rounds to 23,400 cubic feet.

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Question

The contents of a full spherical glass 3 inches in radius are poured into an empty cylindrical glass 6 inches in radius and 6 inches high. What percent of the cylindrical glass is taken up by the contents?

Answer

The volume of the spherical glass is

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

where :

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

$V = \frac{4}{3} \pi \cdot 27$

$V = 36 \pi$

The volume of the cylindrical glass is

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

where :

$V = \pi \cdot 6^{2} \cdot 6$

$V = \pi \cdot 36 \cdot 6$

$V = 216 \pi$

The contents of the spherical glass will take up

$\frac{36 \pi } {216 \pi} \times 100 % = 16 \frac{1}{6} %$

of the capacity of the cylindrical glass.

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Question

The contents of a full cylindrical glass 4 inches in radius and 8 inches high are poured into an empty spherical glass 6 inches in radius. What percent of the spherical glass is taken up by the contents?

Answer

The volume of the cylindrical glass is

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

where :

$V = \pi \cdot 4^{2} \cdot 8$

$V = \pi \cdot 16 \cdot 8$

$V = 128 \pi$

The volume of the spherical glass is

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

where :

$V = \frac{4}{3} \pi \cdot 6^{3}$

$V = 216 \cdot \frac{4}{3} \pi$

$V = 288 \pi$

The contents of the cylindrical glass will take up

$\frac{128 \pi} {288 \pi} \times 100 % = 44 \frac{4}{9} %$

of the capacity of the spherical glass.

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Question

Find the volume of the sphere with a diameter of 5.

Answer

Write the formula for the volume of a sphere.

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

Substitute the radius, which is half the diameter.

$r=\frac{5}{2}$

$V=\frac{4}{3}\pi (\frac{5}{2})^3 = \frac{4}{3}\pi (\frac{5}{2})(\frac{5}{2})(\frac{5}{2})= \frac{1}{3}\pi (\frac{5}{1})(\frac{5}{1})(\frac{5}{2})$

Simplify the terms.

The answer is: $\frac{125}{6}\pi$

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Question

Determine the volume of a sphere with a radius of 5.

Answer

Write the formula for the volume of a sphere.

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

Substitute the radius into the formula.

$V=\frac{4}{3}\pi (5)^3 = \frac{500}{3}\pi$

The answer is: $\frac{500}{3}\pi$

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Question

Determine the volume of a sphere if the diameter is $\sqrt[3]{5}$ .

Answer

Write the formula for the volume of the sphere.

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

Substitute the radius into the formula.

$V=\frac{4}{3}\pi (\sqrt[3]{5})^3$

Any value cubed rooted that is raised to the power of three will leave only the integer behind.

$V=\frac{4}{3}\pi (5) = \frac{20}{3}\pi$

The answer is: $\frac{20}{3}\pi$

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Question

Determine the volume of a sphere with a diameter of 3.

Answer

Given the diameter is 3, the radius is half the diameter, or $\frac{3}{2}$ .

Write the formula for the volume of a circle.

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

Substitute the radius and solve.

$V= \frac{4}{3} \pi (\frac{3}{2})^3=\frac{4}{3} \pi (\frac{27}{8})$

Reduce the fractions.

The answer is: $\frac{9}{2}\pi$

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Question

Determine the volume of a sphere if the diameter is $2\pi^2$ .

Answer

Write the formula for the volume of a sphere.

$V_{sphere} = \frac{4}{3}\pi r^3$

The radius is half the diameter.

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

Substitute the radius.

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

The answer is: $\frac{4}{3}\pi^7$

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Question

Find the volume of a sphere if the radius of the sphere is $3\pi^3$ .

Answer

Write the volume formula for the sphere.

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

Substitute the radius into the formula.

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

Simplify the terms.

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

The answer is: $36\pi^{10}$

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Question

Let $\pi = 3.14$

A sphere has a radius of 3in. Find the volume.

Answer

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

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

where r is the radius of the sphere.

We know $\pi = 3.14$ .

We know the radius of the sphere is 3in.

Now, we can substitute. We get

$V = \frac{4}{3} \cdot 3.14 \cdot (3\text{in})^3$

$V = \frac{4}{3} \cdot 3.14 \cdot 27\text{in}^3$

Now, we can simplify to make things easier. The 3 and the 27 can both be divided by 3. So, we get

$V = \frac{4}{1} \cdot 3.14 \cdot 9\text{in}^3$

$V = 4 \cdot 3.14 \cdot 9\text{in}^3$

$V = 4 \cdot 9\text{in}^3 \cdot 3.14$

$V = 36\text{in}^3 \cdot 3.14$

$V = 113.04\text{in}^3$

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Question

What is the volume of a sphere with a radius of $\frac{3}{2}\pi$ ?

Answer

Write the formula for the volume of a sphere.

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

Substitute the radius into this equation.

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

The answer is: $\frac{9}{2}\pi ^4$

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Question

Determine the volume of a sphere in feet, if the diameter of the sphere is 6 inches.

Answer

Write the formula for the volume of a sphere.

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

We will need to convert the diameter in terms of feet. There are 12 inches to a foot.

$6 \textup{ in} (\frac{1 \textup{ ft}}{12 \textup{ in}}) = \frac{1}{2} \textup{ ft}$

The radius is half the diameter. Multiply this value by one-half to obtain the radius.

$\frac{1}{2} \textup{ ft} \cdot \frac{1}{2} = \frac{1}{4} \textup{ ft}$

Substitute the radius in the formula.

$V= \frac{4}{3}\pi (\frac{1}{4} \textup{ ft})^3 = \frac{4}{3}\pi (\frac{1}{4} \textup{ ft})(\frac{1}{4} \textup{ ft})(\frac{1}{4} \textup{ ft})$

The answer is: $\frac{1}{48}\pi \textup{ ft}^3$

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Question

Let $\pi = 3.14$ .

If a sphere has a radius of 3cm, find the volume.

Answer

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

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

where r is the radius of the sphere.

Now, we know $\pi = 3.14$ . We know the radius of the sphere is 3cm. So, we can substitute. We get

$V = \frac{4}{3} \cdot 3.14 \cdot (3\text{cm})^3$

$V = \frac{4}{3} \cdot 3.14 \cdot 27\text{cm}^3$

$V = \frac{4}{3} \cdot 27\text{cm}^3 \cdot 3.14$

Now, the 3 and the 27 can both be divided by 3. So, we get

$V = \frac{4}{1} \cdot 9\text{cm}^3 \cdot 3.14$

$V = 4 \cdot 9\text{cm}^3 \cdot 3.14$

$V = 36\text{cm}^3 \cdot 3.14$

$V = 113.04\text{cm}^3$

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Question

Find the volume of a sphere with a radius of $\frac{3}{4}\pi$ .

Answer

Write the formula for the volume of a sphere.

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

Substitute the radius into the equation.

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

The answer is: $\frac{9}{16} \pi ^4$

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Question

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

Answer

Write the formula for the volume of a sphere.

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

Substitute the radius.

$V= \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (6) (6) (6) = 288\pi$

The answer is: $288\pi$

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Question

Find the volume of a sphere with a radius of 6cm.

Answer

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

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

where r is the radius of the sphere.

Now, we know the radius of the sphere is 6cm. So, we can substitute. We get

$V = \frac{4}{3} \cdot \pi \cdot (6\text{cm})^3$

$V = \frac{4}{3} \cdot \pi \cdot 216\text{cm}^3$

Now, the 3 and the 216 can both be divided by 3. So, we get

$V= \frac{4}{1} \cdot \pi \cdot 72\text{cm}^3$

$V= 4 \cdot \pi \cdot 72\text{cm}^3$

$V = 4 \cdot 72\text{cm}^3 \cdot \pi$

$V = 288\text{cm}^3 \cdot \pi$

$V = 288\pi \text{ cm}^3$

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Question

Find the volume of a sphere with a surface area of $4\pi$ .

Answer

Write the formulas for the surface area and volume of a sphere.

$A = 4\pi r^2$

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

Substitute the surface area into the first formula and solve for the radius.

$4\pi = 4\pi r^2$

Divide by $4\pi$ on both sides.

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

Square root both sides.

Substitute the radius into the volume formula.

$V=\frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi$

The answer is: $\frac{4}{3} \pi$

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Question

Find the volume of a sphere with a radius of $\pi ^6$ .

Answer

Write the formula for the volume of a sphere.

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

Substitute the radius.

$V=\frac{4}{3}\pi (\pi ^6)^3= \frac{4}{3}\pi (\pi ^6) (\pi ^6) (\pi ^6)$

The answer is: $\frac{4}{3}\pi ^{19}$

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Question

You are 3D printing a ball which will be an integral piece to the project you are creating. If you need the ball to have a radius of 12cm, what volume of material will be required?

Answer

You are 3D printing a ball which will be an integral piece to the project you are creating. If you need the ball to have a radius of 12cm, what volume of material will be required?

We are asked to find the volume of a sphere, to do so, we should use the formula for volume of a sphere.

$V_{sphere}=\frac{4}{3} \pi r^3$

So, let's plug in our radius and solve:

$V_{sphere}=\frac{4}{3} \pi (12cm)^3=\frac{4}{3} \pi 1728cm^3=7234.56 cm^3$

So our answer is:

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Question

You are given a spherical floating speaker for your birthday. If the speaker has a radius of 5 inches, what is its volume?

Answer

You are given a spherical floating speaker for your birthday. If the speaker has a radius of 5 inches, what is its volume?

We need to find the volume of a sphere. To do so, we need the following formula:

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

We know r is 5, so just plug and simplify

$V=\frac{4}{3}\pi (5in)^3\approx 523.56in^3$

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