Thread: Optimization problem - setting up equations

A large soup can is designed to be so taht the can will hold 16pi cubic inches of soup. Find the values of x and h for which the amount of metal needed is as small and possible


x = radius of the cylinder
h = height of the cylinder

I'm having trouble setting up the objective and constraint equations

Obviously, objective is the entire surface area of the cylinder, so:
Area = 2*pi*x^2 + 2*pi*x*h

But for the constraint equation, i can't figure out how to only write it with one variable because you need both in the volume equation which is what they give you 16pi for

vol = 16*pi = pi * x^2 * h

Basically, how can I set these equations up so I can solve them? Thanks!

From the volume equation you have h = 16/x^2. Then you can substitute out h in the area expression and you will be left with a single variable optimization problem (that variable being x).