A farmer decides to build a fence to enclose a rectangular field in which he will plant a crop. he has 1000 feet of fence to use and his goal is to maximize the area of his field.

So a rectangles area is a*b (two sides), its perimater is 2a+2b

We know that 2a+2b=1000 or a+b=500

Nowing this we need to find the max value for a*b

a*b=a * (500-a) = 500a - a^2 = - (a^2-500a)

For simplicity let us work with only a^2-500a

To find the minimum or maximum of a parabola you need to create a perfect square. (like this: (x+y) ^2 - C where C is a constant)

a^2-500a = (a-250) ^2 - 250^2

So this - (a^2-500a)

becomes:

- (a-250) ^2 + 250^2 and you would like to find the max value.

The first part - (a-250) ^2 can be 0 or negative so the max value will be when it is 0.

Thus a=250 - > b=250

This is no big surprise as with given perimeter the biggest area of a rectangal we can get is a square.