1. Draw a sketch. Preferable a sketch of the vertical cross-section.

2. From the volume you can determine the radius of the sphere:

Solve for a.

3. Let r denote the radius of the base circle of the cone and h it's height.

Then

4. According to Euclid's theorem

5. Plug in this term into the equation of the volume to get the equation of a function:

6. Determine the maximum of this function.