Expert: Socrates - 12/30/2009

Question
A juice can in the shape of a right circular cylinder has a fixed capacity. If the material used for the sides of the can costs 0.5 cents/cm^2 and the material for the top and the bottom costs 0.25 cents/cm^2, find the ratio of height to radius that results in a minimum cost.

Answer
Let the height of the can be h and let the radius be r.

To find the cost of the can , we need the area of the top , the area of the bottom and the  area of the side.

The top and bottom of the can are circular disks with radius r . So the area of the top and bottom will each be πr^2.

To find the area of the side , think of what you get if you cut off the top and bottom , cut it down the side and unroll it or flatten it out. You get a rectangle with height h and width equal to the circumference of one of the circles. The circumference will be 2πr , so the rectangle has area 2πrh . This is also the area of the can's side.

Now we can get an expression for the cost of the can.
The cost of the top and bottom is .25 cent per cm^2
The area of the top and bottom will be πr^2 +  πr^2 = 2πr^2
So the total cost of the top and bottom will be (.25)(2πr^2) = .5πr^2

The cost of the side is .5 cent per cm^2
The area of the side is 2πrh
So the total cost of the side is (.5)(2πrh) = πrh

We now know that the cost to make the can will be

.5πr^2 + πrh


Let's suppose the volume of the can is one unit of volume

The volume of the can is also  πr^2 h

so πr^2 h = 1

and then h = 1/πr^2

Substitute 1/πr^2 for h in the expression for cost
and get cost as a function of r

c(r) = .5πr^2 + πr(1/πr^2) = .5πr^2 + 1/r

To find the minimum , set the derivative equal to 0

c'(r) = πr - 1/r^2 = 0

multiply through by r^2

πr^3 - 1 = 0

r = (1/π)^1/3

This is the radius that gives minimum cost.

Since h = 1/πr^2

h = (1/π)^1/3

the height and radius are the same


The ratio h/r is then (1/π)^1/3  /  (1/π)^1/3 = 1

The ratio is 1