example is wrong need help with this Applied Optimization Problem

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 600 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 0.06 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

To minimize the cost of the package:

a) Radius:  ? cm

b) Height:  ? cm

c) Minimum cost ? cents

I keep getting a radius of 3.99295 which is wrong

Answer 1

Surface Area of the sides and bottom = 2pi(r)(h) + pi(r^2)

= pi times the diameter + area of the bottom which is pi times the radius squared

Area of the Top = pi(r^2) = same as the area of the bottom

Volume = pi(r^2)h = 600= area of the bottom times the height

solve for the height in terms of the radius

h = 600/pi(r)^2

substitute into the Surface area equation

area of sides & bottom = 2pi(r)(600/pi(r)^2 + pi(r^2) = 1200/r + pi(r)^2

cost of the top = .04(pi)(r^2

cost of the rest = .06(1200)/r + .06pi(r^2)

total cost = C = 72/r + .06pi(r^2) + .04pi(r^2) = 72/r +.1(pi)(r^2)

C'(r) = -72/r^2 + .2pi(r) = 0

-72 + .2pi(r^3) = 0

r^3 = 72/.2pi = 360/pi

r = cube root of 360/pi = about (360/3.14159)^1/3 = (114.43)^1/3 = about 4.857 cm

radius = 4.857 cm which is less than 2 inches

radius = about 4.9 cm

height = 600/pi(r)^2 = 600/pi(4.857)^2 = about 8.1 cm = about (8.1/2.54 inches)

height = about 8.1 cm

Minimum Cost = 72/r + .1pi(r^2) = 72/4.857 +.1(3.14(4.857)^2 = 14.83 + 7.41 = 22.24 cents

minimum cost = about 22.24 cents

Method is to get Cost in terms of one variable

then take the derivative and set = to zero

solve for that one variable

then solve for the other variable

then plug that into the Cost function

No guarantee that the above has no arithmetic errors or rounding errors, but it's the basic method