A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side x units and a circle of radius of r units. If the sum of the areas of square and the circle so formed is minimum, then

Step-by-step solution

Perimeter of the square =4 x Perimeter of the circle =2 r array ll & 4 x+2 r=2 & 2 x+ r=1 r= 1-2 x array ...(i) Sum of the areas ( A )=x^2+ r ^2 A =x^2+ ( 1-2 x )^2 ...[From (i) A =x^2+ 1 (1-2 x)^2 Differentiating A w.r.t. x , we get aligned & dA ~d x =2 x+ 2 (1-2 x)(-2), d ^2 ~A ~d x^2 =2+ 8 0 & dA ~d x =0 2 x- 4 + 8 x =0 & (2 +8) x=4 & ( +4) x=2 & x= 2 +4 aligned Area is minimum when x= 2 +4 Substituting x= 2 +4 in equation (i), we get aligned & r = 1 +4 & x=2 r aligned