Question

The volume of a sphere is 48 m3. What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter

Answer 1

To find the volume of the cylinder, we first need to determine the radius of the sphere using its volume.

The formula for the volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

where \( r \) is the radius of the sphere. Given that the volume of the sphere is \( 48 \, m^3 \), we can set up the equation:

\[
\frac{4}{3} \pi r^3 = 48
\]

To solve for \( r^3 \), we can rearrange the equation as follows:

\[
r^3 = \frac{48 \cdot 3}{4 \pi} = \frac{144}{4 \pi} = \frac{36}{\pi}
\]

Next, we take the cube root to find \( r \):

\[
r = \sqrt[3]{\frac{36}{\pi}}
\]

Now we can calculate the diameter of the sphere, which is:

\[
d = 2r = 2\sqrt[3]{\frac{36}{\pi}}
\]

Next, we need to find the volume of the cylinder. The volume \( V \) of a cylinder is given by:

\[
V = \pi r^2 h
\]

where \( h \) is the height of the cylinder. Since the height of the cylinder is equal to the diameter of the sphere, we use the diameter we found:

\[
h = d = 2r
\]

Now we substitute this into the volume formula of the cylinder:

\[
V = \pi r^2 (2r) = 2\pi r^3
\]

Substituting \( r^3 \) from earlier:

\[
V = 2\pi \left(\frac{36}{\pi}\right) = 2 \cdot 36 = 72 \, m^3
\]

Thus, the volume of the cylinder is:

\[
\boxed{72 \, m^3}
\]