This relationship between radius, height, and side(slant height) can not exist, at least not in a cone. If your height and radius were 1/3 the side would be the square root of 2/9 or ≈.471

I have tried to follow this thread and your previous posts and have still not figured out exactly what the original problem was. It seems that you have been going along for a while now on this problem.

I think that if you posted a more concise definition of the original problem that you would be pleasantly surprised by a more simple solution. It sounds like some type of basic related rate problem which ususally can be solved quite simply with all of the relevant information from the beginning.

Forgive me if I sound like some type of a-hole, but I think that you somehow made this problem more difficult than it truly needs to be. I, and I suspect others here, would love to help you, but your problem never seemed to be exactly clear from the beginning.