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.