# A gas holder is a flat topped cylinder without a base. Calculate the dimensions of the cylinder...?

A gas holder is a flat topped cylinder without a base. Calculate the dimensions of the cylinder (diameter and height) if it is to contain 1,000,000m^3 of gas and is to be made from a minimum of material. You must show that it is in fact a minimum.

V = pi * r^2 * L = const
A = pi*r^2*L + 2*pi * r * L

Dv = 2*PI * r * L dr + pi * r^2 dL = 0
dr/dl = -r/(2L) : eq A

DA = 2*pi * r * dr + 2 * pi * L * dr + 2 * pi * r dL = 0 for optimization (minimization)
=> dr/dL = -r/(r+L) : eq B
eq A and Eq B =>
2L = r+L => L = R for minimized surface area (to minimize material)

1E6 = pi * r^3 : r = L = 68.27840633 M

r,l,area
68.27840633,68.27840633, 3937.75663, smallest
68.25,68.33525445,43937.76423,...
68.3,68.23523946,43937.76102,L...

The above is given r and calculation of L to make Volume = 1E6 and also area. Note that the surface area is least at the optimum calculation.
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