Question

A cylinder shaped can needs to be constructed to hold 1000 cubic centimeters of soup. The material for the sides
of the can costs 0.1 cents per square centimeter. The material for the top is.2 and the bottom is .3 cents per square centimeter. Find the dimensions for the can that will minimize
production cost.

Answer 1

volume = pi r^2 h =1000
so h = 1000/(pi r^2)

side area = 2 pi r h
top area = pi r^2
bottom area = pi r^2

cost = c=0.1*2 pi r h +(.2+.3)pi r^2
or
c = 0.2 pi r h + .5 pi r^2

c = .2 (1000/r) + .5 pi r^2

dc/dr = 0 at min
= -200/r^2 + pi r
r^3 = 200/pi
r = 4 cm
h =1000/(pi r^2) = 1000/(16 pi) = 20