You are going to make many cylindrical cans. The cans will hold different volume

Question

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 _________.

Explanation / Answer

Area = 2*area of bottom + area of sidewall
A = 2R² + 2Rh
where A is the surface area of the can and R is the radius of the cylinder
Volume = R²h = V, a constant
h = V/R²

A = 2R² + 2R(V/R²)
A = 2R² + 2V/R
dA/dR = 4R - 2V/R² = 0
2R³ = V = R²h
2R = h
Since d²A/dR² = 4 + 6V/R³ > 0, this is a minimum value of A(R).

The minimum surface area for a given volume occurs when 2R = Diameter = h = height.
The ratio of diameter to height that minimizes the area of the cylinder for a given volume = 1

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