Question

A piece of wire 9 m long is cut into two pieces. One piece is bent into the shape of a circle of radius r and the other is bent into a square of side s. How should the wire be cut so that the total area enclosed is:

I need help finding the radius and side of the maximum and minimum. Thanks!

Answer 1

I will go with your definitions

then 2πr + 4s = 9
s = (9-2πr)/4
total area
= A
= πr^2 + s^2
= πr^2 + (9-2πr)^2 /16

dA/dr = 2πr + (1/8)(9-2πr)(-2π)
= 2πr - π(9-2πr)/4
= 0 for a max/min

2πr = π(9-2πr)/4
8πr = π(9-2πr)
8r = 9-2πr
8r + 2πr = 9
r(8+2π) = 9
r = 9/(8+2π) = .6301115.. m
s = 1.26022.. m

so it should be cut at the 4s mark
or at <b>5.04 m for the square, and 3.96 m for the circle</b> to obtain a minimum area
The maximum of course is obtained when all the wire is used for the circle