Question

a piece of wire 24 ft. long is cut into two pieces. one piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. allow x to equal 0 or 24, so all the wire is used for the square or for the circle. How long should each piece of wire be to minimize the total area? what is the radius of the circle? how long is each side of the square?

Answer 1

I will define x in a different way, the way you suggest introduces unnecessary complicated fractions.

The the radius of the circle be r, let each side of the square be x
4x + 2πr = 24
2x + πr = 12
x = 6 - (π/2)r

let A be the total area

A = πr^2 + (6 - (π/2)r)^2
= πr^2 + 36 - 6πr + π^2 r^2/4
dA/dr = 2πr - 6π + π^2 r/2
= 0 for a min of A
divide by π, and multiply by 2
0 = 4r - 12 +πr
12 = r(4+π)
r = 12/(4+π) = appr. 1.68 ft

so 2πr = appr 10.558 ft
leaving (24-10.558)/4 or 3.36 ft for each side of the the square

the wire should be cut so 10.558 is used for the circle and
13.44 ft is used for the square

check:
should get 3.36 by subbing in 1.68 into
x = 6 - πr/2
I get 6 - π(1.68)/2 = 3.36 , yeahhh