Answer:
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
