Question

Volume of Cones, Cylinders, and Spheres Practice
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Question
The volume of a sphere is 72 m3
. What is the volume of a cone if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point)
m3
Remaining Attempts : 3
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Answer 1

To find the volume of the cone based on the given information, we first need to extract the radius of the sphere and then use that to find the volume of the cone.

1. **Volume of a sphere formula**:
\[
V = \frac{4}{3} \pi r^3
\]
Given \( V = 72 \, \text{m}^3 \), we can set up the equation:
\[
\frac{4}{3} \pi r^3 = 72
\]

2. **Solving for \( r \)**:
\[
r^3 = \frac{72 \cdot 3}{4 \pi} = \frac{216}{4 \pi} = \frac{54}{\pi}
\]
\[
r = \left( \frac{54}{\pi} \right)^{\frac{1}{3}}
\]

3. **Finding the diameter of the sphere**:
\[
\text{Diameter} = 2r
\]

4. **Volume of a cone formula**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]
Here, \( h \) (height of the cone) is equal to the diameter of the sphere:
\[
h = 2r
\]

5. **Substituting height and further simplifying**:
\[
V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

6. **Using the volume of the sphere**:
We know from the sphere's volume:
\[
V_{sphere} = \frac{4}{3} \pi r^3 = 72 \Rightarrow r^3 = \frac{72 \cdot 3}{4 \pi}
\]
Thus:
\[
V_{cone} = \frac{2}{3} \pi r^3 = \frac{2}{3} \cdot 72 = 48 \, \text{m}^3
\]

So, the volume of the cone is \( \boxed{48} \, \text{m}^3 \).