Find the dimensions for the can that will minimize production cost.

Question

A cylinder shaped can needs to be constructed to hold 550 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.08 cents per square centimeter.

Find the dimensions for the can that will minimize production cost.

Radius:

Height:

Minimum Cost in cents

Answer 1

Consider any can (not necessarily the optimal) with the desired volume of 550 cm³.

Let

r be the radius of the the can,

h be the height of the can,

V = πr²h = 550 be the volume of the can.

From that

h = 550/(πr²)

The side of the can has area

2πrh = 2πr×550/(πr²) = 1100/r

The top and bottom each have area

πr²

Therefore the cost of manufacturing a can with radius r is

c(r) = 0.03×1100/r + 2×0.08×πr² = 33/r + 0.16πr²

The minimum cost occurs for radius r where the derivative of c(r) is 0

-33/r² + 0.32πr = 0

Solving for r:

r = (33/0.32π)^1/3 = 3.2 cm

Substituting into the formula for h:

h = 550/(πr²) = 17 cm