Design a cylinder can of volume 1000cm^3

Question

Design a cylinder can of volume 1000cm^3 so that it uses the least amount of metal. Show that h=2r, where h is the height of the cylinder and r is its radius.

Answer 1

minimizing metal is simply minimizing surface area.

Area = 2Pir^2+2Pir(h) = (ends plus side)

the volume = 1000 is immaterial if you want to prove h=2r is the best solution

dA/dr = 4Pir + 2Pih

dA/dh = 0 + 2Pir - this implies r = 0. throw this away

4Pir = -2Pih

4r = -2h

-2r=h

in cyclindrical/polar coordinates, a negative radius is the same as a positive radius.