Thread: Optimization - Cylinder inscribed in Cone

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)

i feel like if i can get the relation of h and r i will be good to go
i tried doing this by setting 9/r=21/h and getting r=9h/21 however the computer is telling me that this is wrong