Applied Optimization Problem: A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 600 cubic centimeters of soup....

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 600 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 0.05 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

Helpful information:

h: height of cylinder, 

r: radius of cylinder

Volume of a cylinder: 

Area of the sides: 

Area of the top/bottom: 

To minimize the cost of the package:

Radius: _____ cm

Height: _____ cm

Minimum cost: _____ cent

Answer 1

πr²h = 600 , h = 600/π · r^-2

C = .03 (πr² + 2πrh) + .05πr² = .08πr² + .06πrh

C(r) = .08πr² + 36r^-1

C^'(r) = .16πr - 36r^-2 = 0

r³ = 36/(.16π) ....