A manufacturer wishes to produce the most economical packaging for detergent.

Question

The containers need to hold 25litres and must be cylindrical in shape. What
dimensions will ensure that the least amount of plastic is used to create the
containers. (Note: the volume and surface area of cylinders are V =πr2h and
S = 2πr2 + 2πrh respectively)

Answer 1

v=pi r^2 h = 25, so
h = 25/(pi r^2)

a = 2πr^2 + 2πrh
= 2πr^2 + 2πr*25/(πr^2)
= 2πr^2 + 50/r

da/dr = 4πr - 50/r^2
= (4πr^3 - 50)/r^2

since r≠0, da/dr=0 when 4πr^3 = 50

r = ∛(25 / 2π)