Question

A sphere with a radius of 6 cm has the same volume as a cone with a height of 6 cm. What is the radius of the cone?
Responses
A 2 cm2 cm
B 4 cm4 cm
C 8 cm8 cm
D 12 cm

Answer 1

To solve the problem, we first need to find the volume of the sphere and set it equal to the volume of the cone.

1. **Volume of the sphere**:
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.

For a sphere with a radius of \( 6 \) cm:
\[
V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \text{ cm}^3
\]

2. **Volume of the cone**:
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius of the cone, and \( h \) is the height of the cone.

Given that the height \( h \) of the cone is \( 6 \) cm, we have:
\[
V = \frac{1}{3} \pi r^2 (6) = 2 \pi r^2 \text{ cm}^3
\]

3. **Equating the volumes**:
We set the volume of the sphere equal to the volume of the cone:
\[
288 \pi = 2 \pi r^2
\]

4. **Solving for \( r^2 \)**:
First, we can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[
288 = 2 r^2
\]

Next, divide both sides by \( 2 \):
\[
144 = r^2
\]

Now, take the square root of both sides:
\[
r = \sqrt{144} = 12 \text{ cm}
\]

Thus, the radius of the cone is \( 12 \) cm, which corresponds to option **D**.