Question

A sphere and a cylinder have the same radius and height. The volume of the cylinder is 48 centimeters cubed.

A sphere with height h and radius r. A cylinder with height h and radius r.

What is the volume of the sphere?

Centimeters cubed

Answer 1

To find the volume of the sphere given that the volume of the cylinder is 48 cubic centimeters, we first need to recall the formulas for the volumes of a cylinder and a sphere.

1. **Volume of the cylinder**:
\[
V_{\text{cylinder}} = \pi r^2 h
\]
Since the volume of the cylinder is given as 48 cm³, we have:
\[
\pi r^2 h = 48
\]

2. **Volume of the sphere**:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

Since the cylinder and the sphere have the same radius \( r \) and height \( h \), we can express \( h \) in terms of \( r \) using the volume of the cylinder:
\[
h = \frac{48}{\pi r^2}
\]

However, to find the actual volume of the sphere, we'll note that we need another relationship between \( h \) and \( r \). Since the height of the cylinder is the same as the diameter of the sphere (which is \( 2r \)), we have:
\[
h = 2r
\]

Now, substituting \( h \) back into the volume of the cylinder equation:
\[
\pi r^2 (2r) = 48
\]
\[
2\pi r^3 = 48
\]
\[
\pi r^3 = 24
\]
Now, we can find \( r^3 \):
\[
r^3 = \frac{24}{\pi}
\]

Now, we can find the volume of the sphere using this value of \( r \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Substituting \( r^3 \):
\[
V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{24}{\pi}\right)
\]
\[
V_{\text{sphere}} = \frac{4 \times 24}{3} = \frac{96}{3} = 32
\]

Thus, the volume of the sphere is:
\[
\boxed{32} \text{ centimeters cubed}
\]