If you open up a cylinder into a net then you will observe that it is a rectangle with the top and the buttom being a circle. As the question says the top and the buttom are not painted then this implies that we only consider the side, which is the rectangle.

The area of a rectangle is width multiplied by length. For the cylinder, the length is the height ( ) and the width is the circumference (because this is the length of the circle) this it is .

Therefore, the surface area considered is .

When , the surface area is at which the price is .

Now consider the surface area when , this would equal .

Now we are able to use general arithmetic (Ratio consideration) to solve for the price. When , therefore the price ( ) is

EDIT:I have attached the cylinder diagram and the net considered.