Question

A cylindrical container is to hold 20π cm^3. The top and bottom are made of a material that costs $0.40 per cm^2. The material for the curved side costs $0.32 per cm^2.

Find the height in centimeters of the most economical container.

Answer 1

let the radius be r
let the height be h

V = πr^2 h
πr^2 h = 20π
h = 20/r^2

Cost = .40(2πr^2) + .32(2πrh)
C = .8πr^2 + .64πr(20/r^2)
= .8πr^2 + 12.8π/r
dC/dr = 1.6πr - 12.8π/r^2
= 0 for a min of C

1.6πr = 12.8π/r^2
1.6r^3 = 12.8
r^3 = 8
r = 2
then h = 20/4 = 5

the cylinder should be 5 cm high and have a radius of 2 cm.