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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## Monday, August 11, 2008

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