Consider a wire of length 12 ft. The wire is to be cut into two pieces of length x and 12-x. Suppose the length x is used to form a circle of radius r and the length 12-x is used to form a square with side of length s. What value of x will minimize the sum of their areas?

x = 8

Step-by-step explanation:

When cutting the wire we will get two pieces

x and 12 - x

If we build a circle whith x, the lenght of the circle will be x, and if we look at the equation for a lenght of a cicle 2*π*r = x

then r = x/2π

and consequentely A₁ = area of a circle = πr² A₁ = π*x/2π

A₁ = x/2

With the other piece 12 - x we have to make an square so wehave to divide that piece in four equal length

side of the square = s

s = 1/4 (12 - x) and the area is A₂ = [1/4 (12 - x) ]²

A₂ = (12 - x) ²/16

Then A₁ + A₂ = A (t) and this area as fuction of x

A (x) = x/2 + 1/16 (144 + x² - 24x) A (x) = [ (8x + 144 + x² - 24x) ]/16

Taken derivatives in both sides

A' (x) = 8 + 2x - 24 = 0

2x - 16 = 0 x = 8 and s = 12 - 8 s = 4