Question

Find the dimensions of a rectangle with area 1,000 m2 whose perimeter is as small as possible. (if both values are the same number, enter it into both blanks.)

Answer 1

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)