GED Math - Volume of a Sphere

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

$\frac{4}{3}\pi^7$

$\frac{4}{3}\pi^{10}$

$\frac{4}{3}\pi^6$

$\frac{4}{3}\pi^9$

$\frac{4}{3}\pi^8$

Explanation

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: $\frac{4}{3}\pi^7$

A ball in the shape of a sphere has a radius of inches. What volume of air, in cubic inches, is needed to fully inflate the ball?

Explanation

Recall how to find the volume of a sphere:

$\text{Volume}=\frac{4}{3}\pi r^3$

Plug in the given radius to solve for the volume.

$\text{Volume}=\frac{4}{3}\pi (8^3)=\frac{2048\pi}{3}=2144.66$

Make sure to round to two places after the decimal point.

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

$288\pi$

$144\pi$

$48\pi$

$72\pi$

$324\pi$

Explanation

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: $288\pi$

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 $\pi$ .

$23,400 \textrm{ ft}^{3}$

$22,400 \textrm{ ft}^{3}$

$4,000 \textrm{ ft}^{3}$

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

Explanation

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.

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?

$44 \frac{4}{9} %$

$88 \frac{8}{9} %$

$66 \frac{2}{3} %$

Explanation

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.

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

$\frac{20}{3}\pi$

$200\pi$

$\frac{200}{3}\pi$

$20\pi$

$\textup{The answer is not given.}$

Explanation

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: $\frac{20}{3}\pi$

Let $\pi = 3.14$ .

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

$113.04\text{cm}^3$

$84.78\text{cm}^3$

$28.26\text{cm}^3$

$37.68\text{cm}^3$

$25.12\text{cm}^3$

Explanation

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

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

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

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

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

What is the volume of a sphere if it has a diameter of ?

$V=7776\pi$

$V=23328\pi$

$V=776\pi$

$V=3487\pi$

Not enough information

Explanation

This problem is deceptively simple. In order to solve for the volume of a sphere, all you need is the formula: $V=\frac{4}{3}\pi r^3$ , where r is the radius.

This problem has provided us with the diameter, so we just need to do a little bit of work to solve for the radius. What is the relationship between radius and diameter? The diameter is twice the radius. Or in math speak: . This means we can solve for our radius by taking half of the diameter. Therefore, the radius will be

Now that we have r, we can substitute in the value for r and solve for the volume!

$V=\frac{4}{3}\pi 18^3$

$V=\frac{4}{3}\pi \times 5832$

$V=\frac{5832 \times 4}{3}\pi$

$V=7776\pi$

Consider a tube which is 3 ft wide and 18 ft long.

Find the volume of the largest sphere which could fit within the tube described above.

$V= 4.5\pi ft^3$

$V= 35\pi ft^3$

$V= 14.5\pi ft^3$

Explanation

Consider a tube which is 3 ft wide and 18 ft long.

Find the volume of the largest sphere which could fit within the tube described above.

To find the volume of a sphere, we simply need its radius

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

Now, the largest sphere which will fit within the tube will need to have a radius equal to the tube. Therefore, we can say our radius must be half the diameter, making it 1.5 ft.

Next, plug 1.5 ft into our formula to find our Volume

$V=\frac{4}{3}\pi (1.5ft)^3=4.5\pi ft^3$