If a cone and a sphere have equal radii and have equal volumes, then the ratio between the height of the cone and the diameter of the sphere is:

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `3:2`

Answer

To solve the problem, we need to find the ratio between the height of the cone and the diameter of the sphere given that both have equal radii and equal volumes.

### Step-by-Step Solution:

1. **Define Variables:**
Let the radius of the cone and the sphere be \( r \) (since they are equal). Let the height of the cone be \( h \).

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

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

4. **Set Volumes Equal:**
According to the problem, the volumes of the cone and the sphere are equal:
\[ \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3 \]

5. **Cancel Common Terms:**
We can cancel \( \frac{1}{3} \pi \) from both sides:
\[ r^2 h = 4 r^3 \]

6. **Solve for Height \( h \):**
To isolate \( h \), divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = 4r \]

7. **Find the Diameter of the Sphere:**
The diameter \( d \) of the sphere is twice the radius:
\[ d = 2r \]

8. **Calculate the Ratio:**
We need to find the ratio of the height of the cone to the diameter of the sphere:
\[ \text{Ratio} = \frac{h}{d} = \frac{4r}{2r} \]

9. **Simplify the Ratio:**
Simplifying the ratio gives:
\[ \text{Ratio} = \frac{4}{2} = 2 \]

### Final Answer:
The ratio between the height of the cone and the diameter of the sphere is \( 2:1 \).

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