The solution for this problem is:
Cut wire so first piece has length x.
Second piece has length (72 - x).
Use piece of length x to make circle.
c = circumference, r = radius
c = 2πr = x
r = x/(2π)
A(circle) = πr² = π * (x/(2π))² = x²π/4π² = x²/4π
Use piece of length (72-x) to make square.
s = side length = (72-x)/4
Area(square) = s² = ((72-x)/4)² = (72-x)²/16 = (5184 - 144x + x²)/16
Area(square) = 324 – 9x + x²/16
A = A(circle) + Area(square)
A = x²/4π + 324 - 9x + x²/16
A = x²/4π + x²/16 – 9x + 324
A = 4x²/16π + πx²/16π - 9x + 324
A = (4+π)/16π x² - 9x + 324
This is the function of a parabola that opens up.
To look where A is minimum, you can rewrite equation in vertex form or find
where derivative = 0.
A' = 2(4+π)/16π x - 9 = (4+π)/8π x - 9
A' = 0
(4+π)/8π x - 9 = 0
(4+π)/8π x = 9
x = 9*8π / (4+π)
x ≈ 31.7 inches
