A piece of wire 28 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

a) that part is trivial, the largest area is obtained if all of the wire is used to make a circle

b) let the radius of the circle be r
let the side of the square be x

4x + 2πr = 28
2x + πr = 14
x = (14-πr)/2

A = x^2 + πr^2
= (14-πr)^2 /4 + πr^2

dA/dr = (1/2)(14-πr)(-π) + 2πr = 0 for a max/min
simplifying ...
4r = 14 - πr
4r + πr = 14
r = 14/(4+π)

we need 2πr to form the circle, so the wire should be cut at
2π(14/(4+π)) or appr 12.32 m

check my arithmetic

Answer 2

14/(pi +4)