Applied Maximum and Minimum Problems

Question

A microwaveable cup-of-soup package must be constructed in a cylinder to hold 400 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 glued paper, costing 0.06 cents per square centimeter. Find the dimensions for the package that will minimize production costs.

To minimize the cost of the package:

Radius: ____ cm

Height: ____ cm

Minimum Cost: ____ cents

Answer 1

V = πr²h = 400

h = 400/π · r^-2

bottom area = πr² ; bottom cost = .03πr²

lateral surface area = 2πrh = 2πr(400/π · r^-2) = 800r^-1 ; cost of lateral area = 24r^-1

top area = πr² ; top cost = .06πr²

Total Cost: C(r) = .09πr² + 24r^-1

C^'(r) = .18πr - 24r^-2 = 0

.18r³ - 24 = 0

r = ³√(24/.18) ≈ 5.11 cm

h = 400/π · (5.11)^-2 ≈ 4.88 cm

Min cost: C(5.11) = .09π(5.11)² + 24/5.11 ≈ $12.08