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Thread: Easy but Frustrating Problem

  1. #1
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    Easy but Frustrating Problem

    A circular dartboard is divided into 20 equal sectors, one of which is shown in the diagram. O is the centre of the circle. The areas for scoring double and treble are marked A and B respectively.

    Find the ratio area A:area B in the form n:1, giving n correct to 1 d.p

    OM:99mm
    MN:8mm
    OP:162mm
    PQ:8mm

    Thanks guys
    Attached Thumbnails Attached Thumbnails Easy but Frustrating Problem-dartboard.jpg  
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  2. #2
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    the circle with radius OM has area $\displaystyle \pi 99^2$
    the circle with radius ON has area $\displaystyle \pi (99+8)^2$
    the circle with radius OP has area $\displaystyle \pi 162^2$
    the circle with radius OQ has area $\displaystyle \pi (162+8)^2$

    the sector OM has area $\displaystyle \frac{\pi 99^2}{20}$
    the sector ON has area $\displaystyle \frac{\pi (99+8)^2}{20}$
    the sector OP has area $\displaystyle \frac{\pi 162^2}{20}$
    the sector OQ has area $\displaystyle \frac{\pi (162+8)^2}{20}$
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  3. #3
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    Hello, alibond07!

    You need to know the area of a circle and some common sense.
    Exactly where is your difficulty?


    A circular dartboard is divided into 20 equal sectors,
    one of which is shown in the diagram. O is the centre of the circle.
    The areas for scoring double and treble are marked A and B respectively.

    Find the ratio (area A) : (area B) in the form n : 1, giving n correct to 1 d.p

    OM: 99mm . MN: 8mm . OP: 162mm . PQ: 8mm
    This is a diagram of the radii:
    Code:
          : - - - - 107 - - - :
          : - - -99 - - :
                          (A)             (B)
          O ----------- M --- N ------- P --- Q
    
          : - - - - - - - 162 - - - - - :
          : - - - - - - - - 170 - - - - - - - :

    The circle with radius $\displaystyle OQ = 170$ has area: .$\displaystyle \pi(170^2)$

    The circle with radius $\displaystyle OP = 162$ has area: .$\displaystyle \pi(162^2)$

    . . The area of ring $\displaystyle A$ is: .$\displaystyle 28,\!900\pi - 26,\!244\pi \:=\:2656\pi$


    The circle with radius $\displaystyle ON = 107$ has area: .$\displaystyle \pi(107^2)$

    The circle with radius $\displaystyle OM = 99$ has area: .$\displaystyle \pi(99^2)$

    . . The area of ring $\displaystyle B$ is: .$\displaystyle 11,\!449\pi - 9801\pi \:=\:1648\pi $


    Hence: .$\displaystyle \frac{\text{ring A}}{\text{ring B}} \:=\:\frac{2656\pi}{1648\pi} \;=\;1.611650485$


    Therefore: .$\displaystyle \text{(area A)} : \text{(area B)} \;\approx\;1.6:1$

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