# Thread: Cost of a Can

1. ## Cost of a Can

A can in the shape of a right circular cylinder is required to have a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6 cents per square centimeter, while the sides are made of material that costs 4 cents per square centimeter.

Express the total cost C of the material as a function of the radius r of the cylinder.

2. Let V = the volume of the can.
Let h = the height of the can.

We know that the surface area of the top and bottom of the can is 2 times (pi r^2) - it's 2 times because there's two of them.

The cost of the material is 6 cents per cm^2 so the cost of the top and bottom are 6 x 2 pi r^2 = 12 pi r^2.

The area of the sides is equal to the perimeter times the height, which is 2 pi r h.

The cost of the material is 4 cents per cm^2 so the cost of the sides is 4 x 2 pi r h = 8 pi r h.

We don't know what h is but we do know that V = pi r^2 h, so we can get this in terms of h, that is, h = V / (pi r^2) and then plug that into our equation.

So the total cost is:

12 pi r^2 + 8 pi r (V / pi r^2)

... and we know what V is, it's 500.

So the total cost is 12 pi r^2 + 8 x 500 / r

etc.

Just needs to be simplified a bit and then get the calculator out.

3. ## Good job!

It's great to have people like those on this website helping students with math.