You have 500 ft of fencing all rolled up and you want to make a rectangular playground area for your son. what are the dimensions of the largest playground you could build?

When you read this, the first thing that should jump out at you is: 
What does "largest" mean ? 

Does it mean the longest possible playground ?  The widest possible ? 
The playground with the most possible area ?

Well, we can narrow it down right away.  If you try and find the longest
or the widest possible playground, then what you get is:  The longest or
the widest possible playground is 250 feet by zero.  It has a perimeter
of 500 ft, and nobody can play in it.  That's silly.

It makes a lot more sense if we look for the playground that has
the greatest AREA.

I happen to remember that if you have a certain fixed amount of
fence and you want to use it to enclose the most possible area,
then you should form it into a circle.  And if it has to be a rectangle,
then the next most area will be enclosed when you form it into a square.

So you want to take your 500 feet of fence and make a playground
that's 125-ft long and 125-ft wide.

Its area is (125-ft  x  125-ft)  =  15,625 square feet.

Just to make sure that a square is the right answer, let's test
what we would have if we made it not quite square ... let's say
1 foot longer and 1 foot narrower:

Length = 126 feet
Width  = 124 feet

Perimeter  =  2 (126 + 124)  =  500-ft   good

Area  =    (126-ft  x  124-ft)  =  15,624 square feet.

Do you see what happened ?  We kept the same perimeter, but
as soon as we started to make it not-square, the area started to
decrease.

The square is the rectangle with the most possible area.