# Thread: variant conical glass problem

1. ## variant conical glass problem

A conical glass of depth h and generating angle A, there is carefully dropped spherical iron ball of such size as to cause the greatest overflow. What is the generating angle A that will cause the greates overflow.

2. Originally Posted by pacman
A conical glass of depth h and generating angle A, there is carefully dropped spherical iron ball of such size as to cause the greatest overflow. What is the generating angle A that will cause the greatest overflow.
There is no upper bound to the amount of overflow. Imagine a glass with generating angle very close to 90°. In order to have depth h, it will have to be immensely wide and therefore have an enormous capacity. If a huge ball is (very carefully) lowered into it, the amount of liquid displaced can be arbitrarily large.

If you want a more mathematical argument, notice that the critical value for x found in the previous thread has a denominator which goes to 0 as $A$ approaches 90°. You can check that the same denominator occurs in the expression for the greatest overflow.

3. Thanks opalg, your analysis of the problem is very clear to me, but i am thinking along the line when the ball is totally submerged, barely enough touching the surface of the liquid. That one bothers me . . . ..