Find the largest total surface area (including top and bottom) of a cylinder inscribed in a cone of base radius 9 and height 21.

We need to maximize the following function and :

Then we solve for in terms of obtaining:

We thus obtain the following formula for in terms of alone:

varies over the interval , where

has one stationary point at:

We conclude that the maximum possible total surface area of such a cylinder is:

I know that the surface area is S=2pir(r+h)