Making a cylinder is costing your company $2/in^2 for top and bottom and $6/in^2 for the body(side). If the cylinder is to have 166.25oz volume, what are the most economical cylinder that you can make and what will be the cost of making it.

Answer 1

let radius be r inches
let the height be h inches
vol = πr^2h
166.25 = πr^2 h
h = 166.25/(πr^2)

"economical cylinder" is a function of the surface area, SA

SA = 2 circles + rectangles of the side
cost = 2(2πr^2) + 6(2πrh)
= 4πr^2 + 12πrh
= 4πr^2 + 12πr(166.25/(πr^2))
= 4πr^2 + 1995/r
d(cost)/dr = 8πr - 1995/r^2 = 0 for a max/min of SA
8πr = 1995/r^2
r^3 = 249.375/π
r = appr 4.298 inches
h = appr 2.865 inches

check my arithmetic