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Math Help - Area

  1. #1
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    What percentage of the sector OAB is occupied by the inner circle: 50%, 69%, 75% or 81%

    Could someone please show me a way to find it out?
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  2. #2
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    Quote Originally Posted by xwrathbringerx View Post
    What percentage of the sector OAB is occupied by the inner circle: 50%, 69%, 75% or 81%

    Could someone please show me a way to find it out?
    1. Calculate the radius r of the inner circle: If R is the radius of the quarter circle then

    2r^2 = (R-r)^2~\implies~r^2+2Rr-R^2=0~\implies~\boxed{r=-R+R\cdot \sqrt{2}} The negative solution isn't very plausible here.

    2. The ratio you are looking for is:

    \dfrac{\pi (-R+R\cdot \sqrt{2})^2}{\dfrac14 \pi R^2} = \dfrac{\pi R^2(-1+ \sqrt{2})^2}{\dfrac14 \pi R^2} = 4(1-\sqrt{2})^2 \approx 0.68629...
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