A company wants to build a cylindrical container with a semi-sphere lid.

Question

For a fixed volume V , the company wants to use a minimal amount of material for container
and lid combined. Which radius r and height h of the container minimize the surface area for
container and lid combined?

Answer 1

v = πr^2h + 2/3 πr^3
so, h = (v - 2/3 πr^3)/πr^2
= v/(πr^2) - 2r/3

the surface area is

a = πr^2 + 2πrh + 2πr^2
= 3πr^2 + 2πr(v/(πr^2) - 2r/3)
= 5πr^2/3 + 2v/r

da/dt = 10πr/3 - 2v/r^2
= (10πr^3/3 - 2v)/r^2

da/dt=0 when
10πr^3/3 - 2v = 0
r^3 = 3v/(5π)

and express h in terms of that!

Answer 2

Great, thank you Steve!