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Thread: optimization: maximize area of rectangle inside track

  1. #1

    optimization: maximize area of rectangle inside track

    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?

  2. #2
    Elite Member
    Join Date
    Sep 2005
    Posts
    7,293
    You know the perimeter of the track:

    [tex]\L\\1256=2x+2{\pi}r[/tex]...[1]

    The area of the rectangle is:

    [tex]\L\\2rx[/tex]...[2]


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

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