Question

The volume of a sphere is 36 m^3. What is the volume of a cylinder if it's radius is the same as the sphere's and it's height is equal to the sphere's diameter? Round to the nearest whole number

Answer 1

To find the volume of the cylinder, we first need to understand the relationship between the volume of the sphere and its dimensions.

1. **Volume of the Sphere**:
The formula for the volume \( V \) of a sphere is given by:

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

Given that the volume of the sphere is \( 36 \, \text{m}^3 \), we can set up the equation:

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

To solve for the radius \( r \), we rearrange the equation:

\[
r^3 = \frac{36 \times 3}{4 \pi}
\]

Simplifying this:

\[
r^3 = \frac{108}{4 \pi} = \frac{27}{\pi}
\]

Taking the cube root:

\[
r = \left( \frac{27}{\pi} \right)^{\frac{1}{3}}
\]


2. **Finding the Height of the Cylinder**:
The height \( h \) of the cylinder is equal to the diameter of the sphere. The diameter \( d \) is given by:

\[
d = 2r
\]

Thus, the height of the cylinder is:

\[
h = 2r = 2 \left( \frac{27}{\pi} \right)^{\frac{1}{3}}
\]


3. **Volume of the Cylinder**:
The formula for the volume \( V_c \) of a cylinder is:

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

Substituting \( h = 2r \):

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

We already have \( r^3 \):

\[
V_c = 2 \pi \left( \frac{27}{\pi} \right) = 2 \times 27 = 54 \, \text{m}^3
\]

Thus, the volume of the cylinder is:

\[
\boxed{54 \, \text{m}^3}
\]