A solid sphere of radius `r` is melted and cast into the shape of a solid cone of height `r` , the radius of the base of the cone is

Options

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

Answer

To solve the problem, we need to find the radius of the base of a cone formed by melting a solid sphere of radius \( r \) and casting it into a cone of height \( r \).

### Step-by-Step Solution:

1. **Understand the Problem**: We have a solid sphere of radius \( r \) that is melted and reshaped into a cone with height \( r \). We need to find the radius of the base of the cone.

2. **Write the Volume Formulas**:
- The volume \( V \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi R^2 h \]
where \( R \) is the radius of the base of the cone and \( h \) is its height.

3. **Set the Volumes Equal**: Since the sphere is melted and reshaped into the cone, their volumes are equal:
\[ \frac{4}{3} \pi r^3 = \frac{1}{3} \pi R^2 h \]

4. **Substitute the Height**: We know the height \( h \) of the cone is equal to \( r \):
\[ \frac{4}{3} \pi r^3 = \frac{1}{3} \pi R^2 r \]

5. **Cancel Common Terms**: We can simplify the equation by canceling \( \pi \) and \( \frac{1}{3} \) from both sides:
\[ 4r^3 = R^2 r \]

6. **Further Simplification**: Divide both sides by \( r \) (assuming \( r \neq 0 \)):
\[ 4r^2 = R^2 \]

7. **Solve for \( R \)**: Take the square root of both sides to find \( R \):
\[ R = \sqrt{4r^2} = 2r \]

8. **Conclusion**: The radius of the base of the cone is \( 2r \).

### Final Answer:
The radius of the base of the cone is \( 2r \).

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