Question

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

Answer 1

To find the ratio of the volumes of a cone to a cylinder that share the same radius and height, we first need to use the formulas for the volumes of both shapes.

1. **Volume of a cylinder (V_cylinder)**:
\[
V_{cylinder} = \pi r^2 h
\]

2. **Volume of a cone (V_cone)**:
\[
V_{cone} = \frac{1}{3} \pi r^2 h
\]

Given that the height \( h \) is twice the radius \( r \), we substitute \( h = 2r \) into both formulas.

**For the cylinder**:
\[
V_{cylinder} = \pi r^2 (2r) = 2\pi r^3
\]

**For the cone**:
\[
V_{cone} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

Now, we can determine the ratio of the volume of the cone to the volume of the cylinder:
\[
\text{Ratio} = \frac{V_{cone}}{V_{cylinder}} = \frac{\frac{2}{3} \pi r^3}{2\pi r^3}
\]

The \( \pi r^3 \) terms cancel out:
\[
\text{Ratio} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3}
\]

Thus, the ratio of the volume of the cone to the volume of the cylinder is:
\[
\frac{1}{3}
\]

In summary, the ratio that describes the relationship between the formulas for the volume of a cone to the volume of a cylinder is \( 1:3 \).