The area is: A = x * y = 1000 The perimeter is: P = 2x + 2y The perimeter as a function of x is: P (x) = 2x + 2 (1000 / x) Rewriting: P (x) = 2x + 2000 / x Deriving: P '(x) = 2-2000 / x ^ 2 We match zero: 0 = 2-2000 / x ^ 2 We clear x: 2000 / x ^ 2 = 2 x ^ 2 = 2000/2 = 1000 x = root (1000) x = 10raiz (10) We derive for the second time: P '' (x) = 4000 / x ^ 3 We evaluate x = 10raiz (10) P '' (10raiz (10)) = 4000 / (10 * root (10)) ^ 3 = 0.126491106> 0 (it is a minimum) The dimensions are: x = 10raiz (10) y = 1000 / (10raiz (10)) = 100 / (root (10)) = 100raiz (10) / (root (10) * root (10)) y = 100raiz (10) / (10) y = 10raiz (10) Answer: the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible are: x = 10raiz (10) y = 10raiz (10)
