Question 186475
Let x = width of the pen.


If you have 500 feet of fence, then this is the perimeter, so
2W + 2L = 500


You need to find the length L in terms of x, remembering that x= width.
2x + 2L = 500  You have to solve for L.


Divide both sides by 2 to simplify things a bit:
x + L = 250


Subtract x from each side
L=250-x


So, now you have a formula for Area in terms of x:

A=W*L
A= x*(250-x)


Graph y=x*(250-x)


To find the maximum area, you can graph this equation.  I recommend that you graph this from x=0 to about x=250.  It can't go past these values for x, right?


Now, (did you say that you know what the maximum is?  It's a square, so each side will be 500/4 = 125, and the area will be 125^2 = 15625!!)  graph y values from y=0 up to about 16000.  The graph in this window should look like this:
{{{graph(300,300,-50,250,-500,16000, x*(250-x) ) }}}


This confirms that the maximum occurs at x = 125, and the value there for the area is y = about 15,000 or 16,000.  The exact value can be found by substituting x = 125 into the formula A = x(250-x).


R^2