We have the following categories to consider:

<Aluminum Cost> = (<Area of the top and bottom of the can> + <Lateral area of the can>) * 0.0015

<Label Cost> = (<Area of Label>) * 0.0001

<Attachment cost> = 2 * 0.00125 = $0.0025

The area of ends of the can are each equal to π*5² or 25π. For two ends, that is 50π.

The lateral area of the can is equal to the circumference of the top times the height, or 2 * π * r * h = 2 * 5 * 12 * π = 120π.

Therefore, the total surface area of the aluminum can is 120π + 50π = 170π.  The cost is 170π * 0.0015 = 0.255π, or approximately $0.80.

The area of the label is NOT the same as the lateral area of the can. (Recall that it must be 2 inches longer than the circumference of the can.) Therefore, the area of the label is (2 + 2 * π * 5) * 12 = (2 + 10π) * 12 = 24 + 120π. Multiply this by 0.0001 to get 0.0024 + 0.012π = (approximately) $0.04.

Therefore, the total cost is approximately 0.80 + 0.04 + 0.0025 = $0.8425, or $0.84.

We need to find expressions for the surface area and the volume of a cylinder. The surface area of the cylinder consists of the sum of the surface areas of the two bases plus the lateral surface area.

surface area of cylinder = surface area of bases + lateral surface area

The bases of the cylinder will be two circles with radius r. Thus, the area of each will be πr², and their combined surface area will be 2πr².

The lateral surface area of the cylinder is equal to the circumference of the circular base multiplied by the height. The circumferece of a circle is 2πr, and the height is h, so the lateral area is 2πrh.

surface area of cylinder = 2πr² + 2πrh

Next, we need to find an expression for the volume. The volume of a cylinder is equal to the product of the height and the area of one of the bases. The area of the base is πr², and the height is h, so the volume of the cylinder is πr²h.

volume = πr²h

Then, we must set the volume and surface area expressions equal to one another and solve for r in terms of h.

2πr² + 2πrh = πr²h

First, let's factor out 2πr from the left side.

2πr(r + h) = πr²h

We can divide both sides by π.

2r(r + h) = r²h

We can also divide both sides by r, because the radius cannot equal zero.

2(r + h) = rh

Let's now distribute the 2 on the left side.

2r + 2h = rh

Subtract 2r from both sides to get all the r's on one side.

2h = rh – 2r

rh – 2r = 2h 

Factor out an r from the left side.

r(h – 2) = 2h

Divide both sides by h – 2

r = 2h/(h – 2)

The answer is r = 2h/(h – 2).

When you're calculating the surface area of a cylinder, note that the cylinder will have two circles, one for the top and one for the bottom, and one rectangle that wraps around the "side" of the cylinder (it's helpful to picture peeling the label off a can of soup - it's curved when it's on the can, but really it's a rectangle that has been wrapped around).  You know the area of the circle formula; for the rectangle, note that the height is given to you but the width of the rectangle is one you have to intuit: it's the circumference of the circle, because the entire distance around the circle from one point around and back again is the horizontal distance that the area must cover.

Therefore the surface area of a cylinder =