Minimization of a paper cup

Homework Statement

company manufactures paper cups that are designed to hold 8 fluid ounces
each. The cups are in the shape of a frustum of a right circular cone (so the top and
bottom of the cup are circles, not necessarily of the same size, and the side profile is
that of a trapezoid). What are the dimensions for a paper cup that minimizes the
amount of material used?

Homework Equations

The volume of the cup would be pi/3(R^2+Rr+r^2)h
The surface area would be pi(r)^2+pi(R+r)sqrt((r-R)^2+h^2

The Attempt at a Solution

Know that I'd need to hold volume constant at V=8 fl. oz. or 14.4375 in.^3. Know I need to minimize the surface area of the cup. I solved for h in the volume equation then substituted that into the area equation. I don't know where to go from there because if you take the partial derivatives it becomes way too complicated.

I believe you should use the lagrange multiplier method, minimizing the function of the area using the constant volume function as a constraint. Are you familiar with such thing?

I know sort of how to but not with that function. Also, what constraint would the volume function create

Last edited:
Ray Vickson