Radius of have of a right circuler cone and a sphere is each equal to r. If the sphere and the cone have the same volume. Then what is the height of the cone ?

Options

A. 7r
B. 4r
C. 2r
D. 3r

Answer

To solve the problem, we need to find the height of a right circular cone given that the radius of both the cone and a sphere is equal to \( r \) and that they have the same volume.

### Step-by-Step Solution:

1. **Understand the Volume Formulas**:
- The volume \( V \) of a right circular cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
- The volume \( V \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

2. **Set the Volumes Equal**:
Since the volumes of the cone and the sphere are equal, we can set the two volume formulas equal to each other:
\[ \frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3 \]

3. **Cancel Common Terms**:
We can simplify the equation by canceling out the common terms. The \( \pi \) and \( \frac{1}{3} \) can be canceled from both sides:
\[ r^2 h = 4 r^3 \]

4. **Solve for Height \( h \)**:
To isolate \( h \), divide both sides of the equation by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = \frac{4 r^3}{r^2} \]
Simplifying this gives:
\[ h = 4r \]

5. **Conclusion**:
The height of the cone is \( 4r \).

### Final Answer:
The height of the cone is \( 4r \).