A solid sphere of radius r is melted and recast into the shape of a solid cone of height r. Find radius of the base of the cone.

Answer

To solve the problem of finding the radius of the base of a cone formed by melting a solid sphere, we can follow these steps:

### Step-by-Step Solution:

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

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

3. **Volume of the Cone**: The volume \( V_c \) of a cone is given by the formula:
\[ V_c = \frac{1}{3} \pi R^2 h \]
where \( R \) is the radius of the base of the cone and \( h \) is the height of the cone. Here, we know that \( h = r \).

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

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

6. **Simplify the Equation**: We can divide both sides by \( r \) (assuming \( r \neq 0 \)):
\[ 4r^2 = R^2 \]

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

8. **Final Answer**: The radius of the base of the cone is:
\[ R = 2r \]