text: a wire of length l  is to be cut into two pieces, one being bent to form a square and the other to form a..? .. circle.   how should the wire be cut if the sum of the areas enclosed be the two pieces is to be a minimum let the wire be cut into lengths x and l-x l-x is used to form the square. let each side of the square be s. so 4s = l-x. s = (l-x)/4 area of square = ((l-x)/4)^2 = (l^2-2lx+x^2)/16  x is used to form circle let the radius of circle be r. 2(pi)r = x (circumference of circle) r = x/((2)(pi)) area of circle = (pi)r^2 = x^2/((4)(pi))  now to find the minimum area, you need to differentiate area of square+area of circle and equate it to zero so d/dx((area of square)+(area of circle)) = 0 =>l^2/16-l/8+x/8 +x/((2)(pi))= 0 =>(pi)l^2- 2(pi)l + (pi+4)x = 0  solving the equation using the formula for factorization of quadratic equations.. you will get the value for x in terms of l.