Q) A wire of length l is to be cut into two pieces, one of which is bent to form a circle and the other to form a square. How should the wire be cut so as to minimize the sum of the areas enclosed by the two pieces.

A)
Let the lengths be x and L-x
Area of square = (x/4)2
Area of circle : Given that circumference = L-x = 2.pi.r
r = (L-x) / 2.pi
Area of circle = pi.r^2 = (L-x)^2 / 4.pi
Total Area = x^2 / 16 + (L-x)^2 / 4.pi
Diff and equating to zero:
x/8 = (L-x)/2.pi
pi.x = 4L - 4x
x = 4L/(pi+4)
and thus L-x = L - [4L/(pi+4)] = L.pi / (pi+4)

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