SOLUTION: A rectangular field is to be enclosed by 400 m of fence. What is the maximum area? What dimensions will give the maximum area?

Let x be the length of the rectangle and y be its width.

Then  x + y =  = 200,  and you are asked to find x and y in a way 
to maximize the product  x*y  which is the area.


Express y via x:  y = 200 - x,  and substitute it into the product:

    x*y = x*(200-x).


Next, find the maximum of the quadratic function  f(x) = x*(200-x) = .


  

  
  Now let me remind you that, if you have a quadratic function f(x) =  of the general form,   
    then it reaches the maximum/minimum at x = .
  

       See the lessons
           - HOW TO complete the square to find the minimum/maximum of a quadratic function
           - Briefly on finding the minimum/maximum of a quadratic function
       in this site.


So, in our case the maximum is the vertex at x =  =  = 100.

Thus L = W = 100 meters, and the rectangle is actually a square with the area of 10000 m^2.