Question

A right circular cylinder is inscribed in a cone with height h and base radius r. Find the
largest possible surface area of such a cylinder.

Answer 1

the the radius of the cylinder by x and the height of the cylinder be y.
(the h and r will be constants)

by ratios : x/(h-y) = r/h
y = (hr - hx)/r

I will assume that you want both top and bottom of the cylinder included in your surface area, if not you will have to change the equation.

SA = 2πx^2 +2πxy
= 2πx^2 + 2πhx - (2πh/r)x^2 after subbing in the above y

SA' = 4πx + 2πh - (4πh/r)x
= 0 for a max/min of SA

I get x = hr/(2h-2r)

put that back into SA = ....

I will let you finish the algebra.

(Also check my steps, I tend to make typing errors)