Question

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?

Answer 1

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.