Optimization Prob?

Question

The Can-O-Rad Company Manufactures cylindrical barrels to store nuclear waste. The top and bottom of the barrels are to be made with material that costs $10 per square foot and the rest is made with material that costs $8 per square foot. If each barrel is to hold 5 cubic foot. Find the dimension of the barrel that will minimize the total cost

Answer 1

Volume = pi*r^2*h = 5, so h = 5/(pi*r^2)

Surface Area = top + bottom + lateral area =
pi*r^2 + pi*r^2 + 2pi*rh

Let c = cost. Then
c= 10*(2*pi*r^2) + 8*(2pi*rh)

c= 20pi*r^2 + 16pi*rh

Replace each h with 5/(pi*r^2)

c= 20pi*r^2 + 16pi*r*5/(pi*r^2)

c= 20pi*r^2 + 80r^(-1)

dc/dr = 40pi*r -80r^(-2)

Set this = 0

40pi*r - 80/r^2 = 0

40(pi*r - 2/r^2) = 0

pi*r = 2/r^2

pi*r^3 = 2

r^3 = 2/pi

r = cube root of (2/pi)

Then 5/(pi*(2/pi)^(2/3)