A wire 360 inches long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

Answer 1

let the radius of the circle be r
let the side of the square be x
4x + 2πr = 360
2x + πr = 180 or x = (180- πr)/2

x^2 = πr^2
x = r√π

(180 - πr)/2 = r√π
180 - πr = 2r√π
180 = 2r√π + πr
r(2√π + π) = 180
r = 180/(2√π + π)

so the piece for the circle is 2πr
= 2π(180/(2√π + π) )
= appr 169.1 inches

so the piece for the circle is 169.1 inches and the piece for the square is 190.9 inches

OR

let the piece for the square be a and the piece for the circle be b
a+b = 360

area of square: a^2/16
2πr = b
r = b/(2π)
area of circle = π(b^2/(4π^2)
= b^2/(4π)

b^2/(4π) = a^2/16
b/(2√π) = a/4
4b = 2a√π
but b = 360-a
4(360-a) = 2a√π
1440 - 4a = 2a√π
1440 = a(2√π + 4)
a = 1440/(2√π + 4) = 190.9 yeahhh, same answer
etc