The mathematics problem from my previous post continues to occupy my thoughts. Whilst I have made some progress with a solution, I have come up against a seemingly insuperable barrier. One of the things I need to be able to determine is the ratio into which a section through a frustum of a cone divides the volume, as illustrated above.

I had approached the problem by imagining slicing the cone into discs. The section would then create cicrle segments whose areas could be integrated to obtain the volume. Unfortunately, the expression derived for the area in terms of the height is very hard to integrate. It seems such an obvious question to ask, I can't believe that a solution isn't out there somewhere!