a 1ft long piece of wire needs to be divided into two pieces which will form the shape of a circle and a square. Determine the length of each piece so that the sum of the areas of the circle and the square is minimized

Length of wire for circle is 0.44 ft

Length of wire for square is 0.56 ft

Step-by-step explanation:

Let x is circumference of the circle

then

(1-x) = perimeter of the square

Find the radius of the circle

r = (x/2.π)

Find the area of the circle

a = π (x/2π) ^2

a = π (x^2/4π^2)

Cancel π

a = x^2/4π

Find the area of the square:

a = (1-x/4) ^2

a = (1-2x+x^2) / 16

Total area

A = (x^2/4π) + (1-2x+x^2/16)

A = 4x^2+3.142-6.284x+3.142x^2

A = 7.142x^2-6.284x+3.142

Find the axis of symmetry [x = - b / (2a) ]

x = - (-6.284) / {2. (7.142) }

x = 6.284/14.284

x = 0.44 ft, the piece of wire creating a circle

and

1 - 0.44 = 0.56 ft, the piece of wire creating the square

These lengths should give minimum area of the circle and square together