Question

Volume of Cones, Cylinders, and Spheres Practice
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Question
Suppose a sphere and a cone share the same radius and the height of the cone is twice the radius. Which ratio describes the relationship between the formulas for the volume of a cone to the volume of a sphere?

Answer 1

To find the ratio of the volumes of a cone and a sphere with the same radius, we can start by using the formulas for the volumes of both shapes.

1. **Volume of a sphere**:
\[
V_s = \frac{4}{3} \pi r^3
\]

2. **Volume of a cone**:
\[
V_c = \frac{1}{3} \pi r^2 h
\]
Given that the height of the cone \( h \) is twice the radius \( r \), we can substitute \( h = 2r \) into the formula for the volume of the cone:
\[
V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, we can compare the volumes of the sphere and the cone:

- Volume of the sphere: \( V_s = \frac{4}{3} \pi r^3 \)
- Volume of the cone: \( V_c = \frac{2}{3} \pi r^3 \)

Next, we will calculate the ratio of the volume of the cone to the volume of the sphere:
\[
\text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{2}{3} \pi r^3}{\frac{4}{3} \pi r^3}
\]

Now, we simplify the ratio:
\[
\text{Ratio} = \frac{2}{3} \cdot \frac{3}{4} = \frac{2}{4} = \frac{1}{2}
\]

Thus, the ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{2} \).