Calculus- Derivative

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 550 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.02 cents per square centimeter. The top will be made of glued paper, costing 0.08 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

Answer 1

First, πr²h = 550 so that h=550/πr²

Surface area of can and base = πr² + 2πrh and the cost to make the can and base = .02 * this

Surface area of the top = πr² and the cost to make the top is .08 * this

Substitute the expression for h into the surface area of the can and base.

Then add the two cost expressions to get cost as a function of radius.

Set the derivative of cost = 0 and solve for r.and then get h.

This will be messy!! Good luck.