*PLEASE HELP* Calculus Applied Optimization Problem: A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder....

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of a cylinder to hold 600 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 0.06 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: V = πr^2h

Area of the sides: A = 2πrh

Area of the top/bottom: A = πr^2

To minimize the cost of the package:

Radius:  _____cm

Height:   _____cm

Minimum cost: ____cents

Answer 1

Wow, what fun. Ok, first if πr²h=600, then h=600/πr²

The cost function is C=0.02(2πrh)+0.06(2πr²). Enter what we found h to be, yank the π, and clean up.

C=π[0.04r(600/πr²)+0.12r²]

C=π[24/πr+0.12r²]

Now, put the π back in to clear the fraction of annoyances

C = 24/r + 0.12πr²

Derivative

C' = -24/r² + 0.24πr

C' = 0.24πr-24/r²

The minimum is going to be where that is zero.

0.24πr-24/r² = 0

0.75398r-24/r² = 0

(0.75398r³-24)/r²=0

0.75398(r³-31.8311)/r² = 0

(r³-31.8311)/r² = 0

r³ - 31.8311 = 0 [Legal because we know r>0]

r³ = 31.8311

r= 3.1692

Therefore, r=3.1692" and...

h=600/πr²

h = 600 / 31.5536

h=19.0152

0.04(πrh)+0.12(πr²) = 11.3593 cents per cup. ■