Finding Maximum Area With Gien 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 pronlem is now 'maximise
We can substitute
\[A(x,y)\]to obtain an equation in terms of
\[x\], 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
\[A\]is 5000, when