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 can needs to hold 500 cm^3 of apple juice. The height of the can must be between 6 cm and 15 cm, inclusive. (Assume that there will be no waste.)
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
=> 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.