SOLUTION: Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 600 feet of fencing. (Assume that the length is

  

  

I prepared the Figure on the right to show how I see and understand       
the condition.

In the Figure, L means the length and W means the width of each pen.

So, we have 3 pieces of fencing of the length L each and 4 pieces 
of fencing of the length W each.

Then we have this equation 

3L + 4W = 600,

from which we have W = .
 
  




        Figure. 

 
 
Next, the combined area of the two corals is 

A = L*2W =  =  = ,

and we have to find the length L in a way to maximize the area A, i.e. maximize the quadratic function

A = .

    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 = .

For our situation,  a =   and  b = 300.

Therefore, the maximum is at L = -  =  = 100.

Thus the area get a maximum at L = 100 feet.

Then W =  =  = 75 feet.

Answer.  The area is maximal at L = 100 feet and W = 75 feet.
         Then the area of one coral is 100*75 = 7500 square feet.
         The combined area of the two corals is twice this value, i.e. 15000 square feet.

The plot below confirms this solution. Plot f(L) =