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