A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then 4 π + 1 k is equal to

Step-by-step solution

Let the side of square be x and radius of circle be y . 4 x + 2 π y = 36 (given) Area, A = x 2 + π y 2 ⇒ A = x 2 + π 36 - 4 x 2 π 2 ⇒ ⇒ A = x 2 + 1 π 18 - 2 x 2 ⇒ d A d x = 2 x + 2 π 18 - 2 x - 2 ⇒ d A d x = 2 π x + ( 36 - 4 x ) ( - 2 ) π For critical points d A d x = 0 ⇒ 2 π x - 72 + 8 x = 0 ⇒ x = 36 π + 4 Now, d 2 A d x 2 = 2 + 8 π ⇒ d 2 A d x 2 > 0 Hence, area is minimized. y = 18 π + 4 K [circumferenc