optimization: maximize area of rectangle inside track

[dirtbagbaseball08]

I have this problem i cannot figure out. It uses optimization:

There is a rectangular field with a semicircle on each end and a track around the whole thing. The perimeter of the inner edge of the track is 1256 meters.

I need to maximize the area of the rectangle. How do I do this?

 

[galactus]

You know the perimeter of the track:

\(\displaystyle \L\\1256=2x+2{\pi}r\)...[1]

The area of the rectangle is:

\(\displaystyle \L\\2rx\)...[2]


Solve [1] for x, sub into [2], differentiate, set to 0 and solve for r.