twist on an old optimization problem

We've all seen the classic 'sphere inside a cone', cylinder in a cone', cone in a sphere' problems and etc.

Here's one that is a little different if anyone would enjoy tackling it.

**"Find the ellipsoid of max volume which can be inscribed in a cone of radius R and height H". **

Any willing participants?. The old, cliche, max min and related rates problem have become rather worn out for most of us, I would assume. Though, they are still challenging for calc students. I just thought I would add a twist. Something a little different.