show that the sum of the areas of the circle and the square is leas i

Question

wire of length l is cut in two parts. One part is bent into circle and other into a sqare.  The radius of the circle is half the side of the square.

Answer 1

If the length is cut into two parts a and b then a+b=l.

Let a be the part forming the square, then the side of the square is length a/4 and the radius of the circle is a/8.

The circumference of the circle has length b=2π(a/8)=πa/4. a/b=4/π, l=a+πa/4, therefore a=l/(1+π/4)=4l/(π+4).

The area of the circle=πa²/64 and the area of the square=a²/16.

The total area of the two figures is a²(π+4)/64.

In terms of l this is: (16l²/(π+4)²)((π+4)/64)=l²/(4π+16).