Posted by Anonymous on .
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.

Calculus 
Reiny,
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/(hy) = 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/(2h2r)
put that back into SA = ....
I will let you finish the algebra.
(Also check my steps, I tend to make typing errors)