Finding Maximum Area With Gien Length of Fence

Optimisation problems typically involve answering questions such as 'find the shape that maximises the enclosed area with a given length of fence'.
Suppose we have a 200m of fence to close of part of a field which backs onto a hedge. The hedge and fence are to form a rectangle. This is shown in the diagram, with the green rectangle being the hedge.

The length of fencing is  
The enclosed  
The pronlem is now 'maximise  
subject to  
We can substitute  
  to obtain an equation in terms of  
, which we can then maximise by completing the square.
\[2x+y=200 \rightarrow y=200-2x \rightarrow A(x)=x(200-2x)=200x-2x^2\]
Now complete the square.
\[100-2x^2=-2(x^2-100x)=-2((x-50)^2-50^2)=2 \times 50^2-2(x-50)^2=5000\]
The maximum value of  
  is 5000, when  

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