You are going to make many cylindrical cans. The cans will hold different volumes. But you'd like them all to be such that the amount of sheet metal used for the cans is as small as possible, subject to the can holding the specific volume. How do you choose the ratio of diameter to height of the can? Assume that the thickness of the wall, top, and bottom of the can is everywhere the same, and that you can ignore the material needed for example to join the top to the wall. Put differently, you ask what ratio of diameter to height will minimize the area of a cylinder with a given volume?
That ratio equals
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