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?(1 point)
m3

Answer 1

To solve this problem, we need to first find the radius of the sphere and then use that radius to calculate the volume of the cylinder.

1. **Volume of the Sphere**:
The formula for the volume of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
We are given that the volume \( V \) is 48 m³:
\[
\frac{4}{3} \pi r^3 = 48
\]

To solve for \( r \), rearranging gives:
\[
r^3 = \frac{48 \cdot 3}{4 \pi} = \frac{144}{4 \pi} = \frac{36}{\pi}
\]
Now let's calculate \( r \):
\[
r = \left( \frac{36}{\pi} \right)^{1/3}
\]

2. **Diameter of the Sphere**:
The diameter \( d \) of the sphere is:
\[
d = 2r
\]

3. **Volume of the Cylinder**:
The formula for the volume of a cylinder is:
\[
V = \pi r^2 h
\]
Where \( h \) is the height. From our previous calculation, the height of the cylinder is equal to the diameter of the sphere, so:
\[
h = d = 2r
\]
Therefore, the volume of the cylinder becomes:
\[
V = \pi r^2 (2r) = 2 \pi r^3
\]

4. **Substituting the Volume of the Sphere**:
We know from the first calculation that \( \frac{4}{3} \pi r^3 = 48 \), which means:
\[
r^3 = \frac{36}{\pi}
\]
Now substituting for \( r^3 \) in the cylinder's volume:
\[
V = 2 \pi r^3 = 2 \pi \left( \frac{36}{\pi} \right)
\]

5. **Calculating the Volume of the Cylinder**:
\[
V = 2 \cdot 36 = 72 \, \text{m}^3
\]

Thus, the volume of the cylinder is \( \boxed{72 \, \text{m}^3} \).