Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3?

First, notice that, by the Pythagorean Theorem,

meaning that:

Also, since the volume of a cone with radius r and height h is we know that the volume of the cone is:

Therefore, we want to maximize the function subject to the constraint .

To find the critical points, we differentiate:

Therefore, when

meaning that or . Only is in the interval so that’s the only critical point we need to concern ourselves with.

Now we evaluate  at the critical point and the endpoints:

Therefore, the volume of the largest cone that can be inscribed in a sphere of radius 3 is