A piece of wire 18 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle.

Question

(a) How much wire should be used for the square in order to maximize the total area?

(b) How much wire should be used for the square in order to minimize the total area?

Answer 1

If the square has side s, and the circle has radius r, then the area is
a = s^2 + πr^2
and, since 4s+2πr = 18,
a = s^2 + π((9-2s)/π)^2
since a is a quadratic, it has only a minimum.
da/ds = (2(π+4)s - 36)/π
da/ds=0 when s = 18/(4+π)
so, minimum area when 10 m of wire are used for the square

To find the maximum, find a(0) and a(18/4) and pick the larger value.