You have a wire that is 89 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minimum?

Answer 1

Let the circle have radius r, and the square have side s. Then
2πr + 4s = 89
A = πr^2 + s^2 = πr^2 + (89-2πr)^2
A = (4π^2+π)r^2 - 356πr + 7921
A is minimum when r = 178/(4π+1)
C = 2πr = 356π/(4π+1)