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Math Help - Complicated geometry problem?

  1. #1
    Super Member fardeen_gen's Avatar
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    Complicated geometry problem?

    A point 'O' is situated on a circle of radius R. With centre O, another circle of radius 3R/2 is described. Inside the crescent shaped area intercepted between these circles, a circle of radius R/8 is placed. If the same circle moves in contact with the original circle of radius R, find the length of the arc described by its centre in moving from one extreme position to the other.

    Ans: 7πR/12
    Last edited by fardeen_gen; August 8th 2008 at 09:26 PM. Reason: Lots of typos, wrote centroid instead of contact...
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  2. #2
    MHF Contributor
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    I'm pretty sure this one will require a MUCH better description, or a drawing.
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  3. #3
    Super Member fardeen_gen's Avatar
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    @ TKHunny: edited
    Added an image. But I couldnt figure out where to put the circle with radius R/8. Can someone help me figure that out?
    Attached Thumbnails Attached Thumbnails Complicated geometry problem?-math.bmp  
    Last edited by fardeen_gen; August 8th 2008 at 11:25 PM.
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  4. #4
    Super Member wingless's Avatar
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    I think the question wants you to find the lenght of the arc in the shape.

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  5. #5
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    earboth's Avatar
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    Quote Originally Posted by wingless View Post
    I think the question wants you to find the lenght of the arc in the shape.
    Quote Originally Posted by fardeen_gen View Post
    ... If the same circle moves in contact with the original circle of radius R, find the length of the arc described by its centre in moving from one extreme position to the other.

    Ans: 7πR/12
    I don't want to pick at you but as far as I understand the problem the small circle moves along the original circle.

    I have the greatest problems to determine the "extreme positions"
    Attached Thumbnails Attached Thumbnails Complicated geometry problem?-abrollkreis1.gif  
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  6. #6
    Super Member wingless's Avatar
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    Quote Originally Posted by earboth View Post
    I don't want to pick at you but as far as I understand the problem the small circle moves along the original circle.

    I have the greatest problems to determine the "extreme positions"
    You're right
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  7. #7
    Super Member wingless's Avatar
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    I found the answer using earboth's shape, the answer appears to be \frac{\pi R}{80} \arccos \frac{1}{6}. I also checked fardeen's answer, but it doesn't agree with the numerical approximation.

    So, using the answer, I found the question to be,



    The question is trivial after you find the angle in this shape:



    And you can easily find it using law of ..... =)
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