Question

A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals of the same dimensions. What dimensions produce a maximum enclosed area? [Use complete-the-square to solve.]

Answer 1

Answer:

  50 ft by 33 1/3 ft

Step-by-step explanation:

Let x represent the dimension perpendicular to the shared fence. Then the dimension parallel to the shared fence will be ...

  y = (200-2x)/3

The area will be the product of these dimensions, so will be ...

  area = xy = x(200 -2x)/3 = (2/3)x(100 -x)

  area = (-2/3)(x^2 -100x)

To complete the square, we need to add the square of half the x-coefficient inside parentheses, and the opposite of that quantity outside parentheses.

  area = (-2/3)(x^2 -100x +2500) +(2/3)(2500) . . . . complete the square

  area = (-2/3)(x -50)^2 + 5000/3

The vertex of this parabolic curve is at x=50, so the dimensions of the maximum area are ...

  x = 50 . . . feet

  y = (200 -2·50)/3 = 33 1/3 . . . feet