Chapter 9, Problem 10P

A paper cup shaped as a frustum of a cone with $R_2=2R_1$ , is designed to have a volume of 250 cm³. Determine R₁, R₂, and h such that the least amount of paper will be used for making the cup.

The volume and the surface area of the paper cup are given by:

$V=\frac{1}{3}\pi h\left(R_1^2+R_2^2+R_1{R}_2\right)$

$S=\pi \left(R_1+R_2\right)\sqrt{{\left(R_2-R_1\right)}^2+h^2}+\pi \left(R_1^2+R_2^2\right)$