This is taking the classic 'minimizing the surface area of a cylinder' problem one step further. In the first part of the problem, it was given that the volume is 1000mL. I found the height and radius that would minimize the surface area.
The second part of the problem is this:
The material for the cans is cut from sheets of metal. The cylindrical sides are
formed by bending rectangles; these rectangles are cut from the sheet with little or no waste. But if
the top and bottom discs are cut from squares of side 2r, this leaves considerable waste metal, which
may be recycled but has little or no value to the can makers. If this is the case, show that the total
amount of metal used (including the waste metal) is minimized when h/r is 8/(pi)
The task is to show that the ratio of height to radius is 8pi).
How is this problem done?