Calculus I Applied Optimization Problem (Objective: Find the dimensions for the package that will minimize production cost.)

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of 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 cost.

To minimize the cost of the package:

Radius: ___ cm

Height:  ___  cm

Minimum cost: ___  cents

Answer 1

Volume, V= πr²h = 250

h = 250/(πr²)

Cost, C= 0.03(BottomArea + SideArea) + 0.08(TopArea)

C= 0.03(πr²+2πrh)+0.08(πr²h)

Subtituting for h

C(r) = 0.03πr²+0.06πr(250/(πr²) + 0.08πr² = 0.11πr² + 15/r

C'(r) = 0.22πr - 15/r²

Setting C'(r) to zero and solving for r

r³=15/(0.22π)

h = 250/(π(15/(0.22π)^1/3)

C((15/(0.22π)^1/3) ≈ 8 cents