What would be the maximum possible area of a rectangular field...

Question

Maximum Area: What would be the maximum possible area of a rectangular field enclosed by 500-meter fencing?

Answer 1

Let x and y be the dimensions of the rectangular field. If the perimeter, p, of a rectangle is 500 meters.

p = 2x + 2y = 500

y=(500-2x)/2 = 250-x

Let A be the area of the rectangle.

A = xy = x(250-x) = 250x-x²

A(x) = 250x-x²

This is the equation of an inverted parabola. The vertex of the parabola gives the maximum value. The vertex of a parabola can be found using

x=-b/2a

where the parabolic equation is: ax²+bx+c. Here, a = -1 and b = 250, so the vertex is:

x = -(250/(2(-1)) = 125.

A(125)= 250(125)-(125)² = 31250-15625 = 15625 m²