a piece of wire 12 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 12, 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?

Question

Reiny · Accepted Answer

Let the radius of the circle be r and each side of the square be x
so 2πr + 4x = 12
πr + 2x = 6
x = (6-πr)/2

Area = πr^2 + x^2
= πr^2 + ((6-πr)/2)^2
= πr^2 + (3 - πr/2)^2

d(Area)/dr = 2πr + 2(3-πr/2) (-π/2)
= 2πr - π(3-πr/2)
= 0 for a minimus Area

2πr = π(3-πr/2)
2r = 3-πr/2
4r = 6 - πr
4r+πr = 6
r = 6/(4+π) = appr .840

so we need 2πr or 5.2788 ft for the circle,
leaving 6.721 ft for the square.

radius of circle = .840 ft
side of each square = 1.68 ft

check:
2πr + 4x
= 2π(.840) + 4(1.68) = 11.998 , not bad

let r = .8 , then x = 1.743
area = π(.8)^2 + 1.743^2 = 5.049
let r = .9, then x = 1.586
area = π(.9)^2 + (1.586)^2 = 5.06.

our answer
r = .84 , x = 1.68
area = π(.84)^ + 1.68^2 = 5.039 , which is lower than either of the
slightly larger and slightly smaller radii.