Applied Optimization Problems need help

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 400 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.07 cents per square centimeter. Find the dimensions for the package that will minimize production cost

To minimize the cost of the package:

Radius ?

Height ?

Minimum cost ?

Answer 1

Cost function C(r,h) = (πr² + 2πrh)·0.04 + πr²·0.07 = 0.11πr² + 0.08πrh; but πr²h = 400; h = 400/(πr²)

So, C(r) = 0.11πr² + 0.08πr·400/(πr²); C(r) = 0.11πr² + 32/r;

C'(r) = 0.22πr - 32/r² = 0; 0.22r³π = 32; r = (32/(0.22π))^1/3 = 3.591 cm

C''(r) = 0.22π + 64/r³; C''(3.591) = 0.22π + 64/3.591³ = 2.073 > 0. So, we have minimum

h = 400/(π·3.591²) = 9.874 cm

Minimum cost C_min = 0.11π·3.591² + 32/3.591 = 13.37centc

Answer 2

Here is a tentative solution assuming the costs to be $0.07 and $0.04. Please check for accuracy.

If the cylinder height is h and radius is r, then:

1. The cylinder surface area is given by: A=2πrh + 2πr²
2. The volume of the cylinder is given by: V=πr²h = 400 or h= 400/πr²
3. We may break down the total cost of the finished cylinder as equal to the components (sides, bottom, top) of the total surface area:

C= 0.07*2πrh + 0.07*πr² + 0.04*πr² =

0.07*2πr*(400/πr²) + 0.07*πr² + 0.04*πr²

or

C = (56/r ) + 0.11*πr²

Now we can set the derivative of the cost function (w.r.t. r) to find its minimum:

dc/dr = (-56/r² ) + 0.22*πr = 0

Multiplying both sides by r² we obtain:

r³= 56/(0.22 π) or r = 27.0081 cm.

Now we may use the estimated radius and calculate h and minimum cost as:

h= 0.174551 cm and C=$254.14.