In the following problem, how should the can be constructed so that a minimum amount of material will be used in the construction? A cylindrical...

The volume of a cylindrical container is pi*r^2*h where h
is the height of the cylinder and the radius of the base is
r.

The required can should have a capacity of 500
cm^3.

=> 500 =
pi*r^2*h

=> h =
500/pi*r^2

The surface area of the container is 2*pi*r*h +
2*pi*r^2

Eliminating h we
get:

Area = 2*pi*r*500/pi*r^2 +
2*pi*r^2

=> Area = 1000/r +
2*pi*r^2

Now Area has to be minimized. Take the derivative
of Area and solve for r.

=> -1000/r^2 + 4*pi*r =
0

=> -1000 + 4*pi*r^3 =
0

=> r =
(250/pi)^(1/3)

=> r = 4.30
cm

Height =
500/pi*r^2

=>
500/pi*(4.3)^2

=> 8.6
cm

The height of the cylinder
to minimize the material required to construct it and have
a volume of 500 cm^3 should be 8.6 cm.