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.
