How can a right circular cylinder be designed to hold 22 cubic inches of a soft drink?

Question

The cost for the material for the top and bottom of the can is twice the cost for the material of the sides.
Let r represent the radius and h the height of the cylinder.

Answer 1

The volume of the can will be height h times surface of bottom.
Hence V = h pi r^2.
As this equals 22 cubic inches we find h= 22/(pi r^2).

The cost of the can equals side area + 2 times (bottom + top).
Hence C = 2 pi r h + 4 pi r^2.

Substitute for h: C= 4 pi r^2 + 44 pi r/(pi r^2) = 4 pi r + 44 /r

To find the minimal cost we take the derivative and put it to zero.

dC/dr = 8 pi r - 44 / r^2 = 0 or r^3 = 5.5/pi or r=1.21 inch

Answer 2

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