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Math Help - circl trig problem

  1. #1
    Member smmmc's Avatar
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    circl trig problem

    two circles of radii 3 cm and 4dm have their centres 5 cm apart. calculate the area of the region common to both circles.

    Formula i use is 1/2r^2(theta - sintheta)

    my digram below.. not sure if its correct.. also what angles to i have to find ?


    thanks
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  2. #2
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    Quote Originally Posted by smmmc View Post
    two circles of radii 3 cm and 4dm have their centres 5 cm apart. calculate the area of the region common to both circles.

    Formula i use is 1/2r^2(theta - sintheta)

    my digram below.. not sure if its correct.. also what angles to i have to find ?


    thanks
    The common area consists of two lens-shaped areas.
    One lens-shaped area is the difference of the area of a sector and an isosceles triangle.

    The triangle CPM is a 3-4-5 right triangle. Therefore

    \sin\left(\dfrac{\mu}2\right)=\dfrac35~\implies~\m  u \approx 73.74^\circ

    Then the area of the left lens-shaped area is:

    A=\underbrace{\dfrac{\mu}{360^\circ} \cdot \pi \cdot 4^2}_{sector} - \underbrace{\dfrac12 \cdot 4^2 \cdot \sin(\mu)}_{isosceles\ triangle} \approx 2.616

    To calculate the second part of the common region is left for you.
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