The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.

Explanation

To solve the problem, we need to show that when the sum of the volumes of a sphere and a cube is minimized, the diameter of the sphere is equal to the edge of the cube. Let's go through the steps systematically.

### Step 1: Define the Variables
Let:
- \( r \) = radius of the sphere
- \( a \) = edge length of the cube
- \( S \) = sum of the surface areas of the sphere and cube
- \( V \) = sum of the volumes of the sphere and cube
...