The height of a cone is equal to the radius of its base. The radius of a sphere is equal to the radius of the base of the cone. The ratio of the volume of the cone to the volume of the sphere is

Options

A. `(1)/(12)`
B. `(1)/(4)`
C. `(1)/(3)`
D. `1/1`

Explanation

To solve the problem, we need to find the ratio of the volume of a cone to the volume of a sphere, given that the height of the cone is equal to the radius of its base, and the radius of the sphere is equal to the radius of the base of the cone.

### Step-by-Step Solution:

1. **Define Variables**:
Let the radius of the base of the cone (and the radius of the sphere) be \( r \). Since the height of the cone is equal to the radius of its base, the height \( h \) of the cone is also \( r \).

2. **Volume of the Cone**:
The formula for the volume \( V_c \) of a cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h \]
Substituting \( h = r \):
\[ V_c = \frac{1}{3} \pi r^2 r = \frac{1}{3} \pi r^3 \]

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

4. **Finding the Ratio**:
We need to find the ratio of the volume of the cone to the volume of the sphere:
\[ \text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{1}{3} \pi r^3}{\frac{4}{3} \pi r^3} \]
Here, \( \pi r^3 \) cancels out:
\[ \text{Ratio} = \frac{1/3}{4/3} = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4} \]

5. **Conclusion**:
The ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{4} \).

### Final Answer:
The ratio of the volume of the cone to the volume of the sphere is \( \frac{1}{4} \).