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

Question

3.5

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 550 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: V = pir^2h Area of the sides: A = 2pirh Area of the top/bottom: A = pir^2 To minimize the cost of the package:

Explanation / Answer

Given, h : height of cylinder,
r : radius of cylinder
Volume of a cylinder: V=r2h
Area of the sides: A=2rh
Area of the top/bottom: A=r2

The cost=C= C = .03(2pi*rh) + .03(pi*r^2) + .05(pi*r^2)
C = .03(2pi*rh) + .08(pi*r^2)
Given, V = pi*r^2h = 550
=> h = 550 /(pi*r^2)
C = .03(2pi*r*550 /pi*r^2) + .08pi*r^2
=> C = 33*r^(-1) + .08pi*r^2
=> C'=dC/dr = -33*r^(-2) + 0.16pi*r
For Minimum ocst C'=0 =>33 = 0.16pi*r^3 =>r^3=33/0.16=206.25
=>r= (206)^(1/3) = 5.90
therefore h=550/(pi*5.90^2)=5.03
Therfore C=33*r^(-1) + .08pi*r^2=12.99 cents

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