a wire of 30-x lenght made by it square so what is the premeter of square ? and why ?

Question

a wire of length 30 cm .id devided into two parts,the lenght of one of the two parts is x cm. this part is turned to be circle and other part is turned to be square . if sum of the surface areas of the circle and the square is minimum then find x

Answer 1

The circle has a circumference of x, so its radius r is given by x=2πr, r=x/(2π).

Its area is πr²=π(x²/(4π²)=x²/(4π).

The side of the square=(30-x)/4.

Its area is ((30-x)/4)²=(30-x)²/16.

Total area of square and circle A=x²/(4π)+(30-x)²/16 (parabola).

A=(16x²+4π(30-x)²)/(64π),

A=(4x²+900π-60πx+πx²)/(16π),

A=(4+π)(x²-60πx/(4+π)+900π/(4+π))/(16π).

Completing the square for x²-60πx/(4+π)+900π/(4+π):

x²-60πx/(4+π)+900π²/(4+π)²-900π²/(4+π)²+900π/(4+π)=

(x-30π/(4+π))²+3600π/(4+π)². 3600π/(4+π)² is a positive quantity.

When this square=0 we have a minimum value for the area, and x=30π/(4+π). Other values of x give a positive perfect square which would increase the area because a square is always positive.

x=13.197 approx. [Area=31.506 approx=225/(4+π)].

The side of the square=30-13.197=16.803 approx so the perimeter of the square is 4 times this=67.212 approx.