A sphere of radius r has the same volume as that of a cone with a circular base of radius r. Find the height of the cone.

Options

A. 2r
B. `r/3`
C. `4r`
D. `((2)/(3))r`

Answer

To solve the problem, we need to find the height of a cone given that its volume is equal to the volume of a sphere, both having the same radius \( r \).

### Step-by-step Solution:

1. **Write the formula for the volume of a sphere**:
\[ V_{sphere} = \frac{4}{3} \pi r^3 \]

2. **Write the formula for the volume of a cone**:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

3. **Set the volumes equal to each other**:
Since the volume of the sphere is equal to the volume of the cone, we can write:
\[ \frac{4}{3} \pi r^3 = \frac{1}{3} \pi r^2 h \]

4. **Cancel out common terms**:
We can cancel \( \pi \) from both sides and \( \frac{1}{3} \):
\[ 4 r^3 = r^2 h \]

5. **Rearrange the equation to solve for \( h \)**:
To isolate \( h \), we can divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = \frac{4 r^3}{r^2} \]

6. **Simplify the expression**:
Simplifying the right-hand side gives:
\[ h = 4r \]

7. **Conclusion**:
Therefore, the height of the cone is:
\[ h = 4r \text{ cm} \]

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

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