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Math Help - Cool Geometry Question

  1. #1
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    Cool Geometry Question

    Ok this looks easy, but it isn't. To solve it, you need to obtain three independent equations containing x,y and z that can be solved to determine the areas of the regions.

    A square has sides 6 cm long. Four quarter circles are inscribed in the square. Determine the areas of the three different kinds of regions that are formed.

    Here's the picture:



    So you need to make 3 equations with three unknown then solve. I happen to know two of them. They are:

    4x + 4y + z = 36

    This is because there are 4 "x" regions, 4 "y" regions, and 1 "z" region making up the square with area 6^2, aka 36.

    2x+3y+z = 9pi

    this is because there are 2 "x" regions, 3 "y" regions, and 1 "z" region making up a quarter circle with area (r^2 pi)/4 = (36pi)/4 = 9pi

    What is the third equation? Its more difficult to find than the first two, I believe. Make sure your answer isn't just a combination of the first two, like 2x + y = 36 - 9pi

    Thank you for all the help you can offer me.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chris.j.stanford View Post
    Ok this looks easy, but it isn't. To solve it, you need to obtain three independent equations containing x,y and z that can be solved to determine the areas of the regions.

    A square has sides 6 cm long. Four quarter circles are inscribed in the square. Determine the areas of the three different kinds of regions that are formed.

    Here's the picture:



    So you need to make 3 equations with three unknown then solve. I happen to know two of them. They are:

    4x + 4y + z = 36

    This is because there are 4 "x" regions, 4 "y" regions, and 1 "z" region making up the square with area 6^2, aka 36.

    2x+3y+z = 9pi

    this is because there are 2 "x" regions, 3 "y" regions, and 1 "z" region making up a quarter circle with area (r^2 pi)/4 = (36pi)/4 = 9pi

    What is the third equation? Its more difficult to find than the first two, I believe. Make sure your answer isn't just a combination of the first two, like 2x + y = 36 - 9pi

    Thank you for all the help you can offer me.
    There is some kind of problem with the diagram, I can't see it.

    RonL
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  3. #3
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    Link fix

    Try this link

    New Page 1
    Last edited by chris.j.stanford; April 26th 2007 at 03:40 AM. Reason: fixed the link
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  4. #4
    Grand Panjandrum
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    Area of square:

    36 = z+4x+4y

    Area of a quarter circle:

    9 pi = z+2x+3y

    Area of central lense shape:

    2[18(pi/2-1)] = z+2y

    You have three simultaneous equations in three unknows which you should be able to solve.

    Except checking these are not linearly independent! So we still need another equation.

    RonL
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  5. #5
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    Hmmm

    Yes, that problem is run into alot, the third equation is always sneakily just a combination of the other two. Thanks for trying, anyways... My teacher says it has something to do with symmetry.
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