A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

Step-by-step solution

Minimizing the sum of areas formed by bending a wire of length L = 8 into a square and circle. Let x be the length used for the square, so the circle uses 8 - x . The square's perimeter is x , giving side length s = x 4 and area A_s = x^2 16 . The circle's circumference is 8 - x , so its radius is r = 8-x 2 and area A_c = (8-x)^2 4 . The total area function is A(x) = x^2 16 + (8-x)^2 4 . Differentiating to find the minimum: dA dx = x 8 - 8-x 2 Setting the derivative to zero: x 8 = 8-x 2 2 x = 64 - 8x x(2 + 8) = 64