Question

Prove that the radius of the base of right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone

Answer 1

Hmm. If the cone has radius R and height H, and the cylinder has radius r and height h,

by similar triangles, r/R = (H-h)/H

area of cylinder
a = 2pi r h
= 2pi (R/H) h(H-h)
da/dh = 2pi (R/H) (H-2h)
max a when H = 2h

or, if you don't have calculus to help you, let's pick up here:

a = 2pi (R/H) h(H-h)
= 2pi (R/H) (Hh - h^2)
this is a parabola with vertex (maximum value) when h = H/2