Question

A microwaveable cup-of-soup package needs to be constructed in the shape of a cylinder to hold 250 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.03 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 costs.

To minimize the cost of the package:
Radius =?
Height=?
Minimum cost=?

Answer 1

I will skip the units for easier typing

let the radius be r, let the height be h
Need: V = 250
πr^2 h = 250
h = 250/(πr^2)

cost = .03(πr^2) + .03(2πrh) + .08(πr^2)
= .11π r^2 + .06πr(250/(πr^2)
= .11πr^2 + 27.5/r

d(cost)/dr = .22πr - 27.5/r^2 = 0 for a min of cost

.22πr = 27.5/r^2
r^3 = 27.5/(.22π) = 39.7887..
r = 3.4139.. cm
h = 6.8278.. cm , (notice h = 2r)

sub those into the cost equation to find the min cost
Your answer will be in cents.