Q:

Optimisation: volume of a cone

Question: A right triangle with hypotenuse of length a is rotated about one of its legs to generate a right circular cone. Find the greatest possible volume of such a cone.
My thoughts:
Given the formula for the volume of a cone and the Pythagorean Theorem, I can eliminate either the base or the height variable from the formula for the volume of a cone, giving me volume in terms of hypotenuse and height or volume in terms of hypotenuse and base.
I think that the only way to determine the greatest possible volume of such a cone is to assume that the length of the hypotenuse (a) is constant, as I will otherwise be unable to find the derivative (dV/dh or dV/db) and then determine the maximum.
Is this the case? Should I treat the length of hypotenuse as a constant?

A:

YES.
You're told you have 'a right triangle with hypotenuse of length $a$'.
That means the length is constant. The value is not specified now and it's hidden behind the symbol $a$, but it is some constant.

