Question

You want to make cylindrical containers of a given volume V using the least amount of construction material. The V

side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top
and bottom are cut from squares of side 2r, so that 2(2r)^2 = 8r^2 of material is needed (rather than 2πr^2, which is the
total area of the top and bottom). Find the optimal ratio of height to radius.​

Answer 1

8/π

Explanation:

Volume of the cylinder is:

V = πr²h

Surface area of the cylinder (with square top and bottom) is:

A = 2πrh + 8r²

Use the volume equation to write h in terms of r.

h = V / (πr²)

Substitute into the area equation:

A = 2V / r + 8r²

Take derivative with respect to r.

dA/dr = -2V / r² + 16r

Set to 0 and solve for r.

0 = -2V / r² + 16r

2V / r² = 16r

2V = 16r³

V = 8r³

Plug into the volume equation.

8r³ = πr²h

8r = πh

h / r = 8 / π