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Thread: Frustrating area calculation

  1. #1
    May 2011

    Frustrating area calculation

    First of all, hello, I'm new, and I make awful sketches on the computer. Now, for the problem.

    This is roughly the sketch I was given (they're supposed to be perfect quarter circles going from the edges of the square to the opposite one). The length of a side is 1. What I need to know is the middle part Z's area (and then find the percentage of it compared to the full square). I've been sitting for over a week trying to figure this out, but I'm going in circles and it's really starting to confuse me.

    What I have so far (not that much):
    3x + 2y + z = π/4
    x + 2y = 1 - (π/4)
    2x + z = (π/2) - 1
    x - 2y + z = (3π/4) - 2
    4y - z = π - 3

    After this, I am incredibly confused. I've tried everything I know to solve this, but to no avail. I'm betting the answer really is super simple and I just can't find it...

    Thanks on beforehand for help!

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  2. #2
    MHF Contributor
    Oct 2009
    You have three variables and two independent equations:

    3x + 2y + z = π/4
    4x + 4y + z = 1

    To have a unique solution, you need three equations. My guess is that you can't form the third equation without using the facts that these are actual circles. It may be possible to twist the shape a little preserving the two equations above.

    Anyway, you an find y/2 using trigonometry. That would be the third equation.
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