# 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...

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.)

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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. **