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Math Help - More circles, arcs and sectors

  1. #1
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    More circles, arcs and sectors

    I don't usually struggle with maths, but if anybody could help me, it would be appreciated again!

    The length of an arc of a circle is 12cm. The corresponding sector area is 108cm squared. Find:
    a) the radius of the circle.
    b) the angle subtended at the centre of the circle by the arc.

    See attached and check if I have drawn the correct diagram.

    Help is much appreciated again,
    Thanks
    D Loizou
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  2. #2
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    s = r\theta

    A = \frac{r^2 \theta}{2} = \frac{r(r\theta)}{2}

    you are given s and A ... see the substitution you need to do?
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  3. #3
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    Hello, loizoud94!

    The length of an arc of a circle is 12 cm.
    The corresponding sector area is 108 cm².

    Find:
    a) the radius of the circle.
    b) the angle subtended at the centre of the circle by the arc.
    Code:
                  * * *    A
              *           *
            *           /   *
           *         r /     *
                      /
          *          / θ      *
          *         * - - - - * B
                    O    r

    Length of arc: . s \:=\:r\theta

    . . We have: . r\theta \:=\:12 .[1]


    Area of sector: . A \:=\:\tfrac{1}{2}r^2\theta

    . . We have: . \tfrac{1}{2}r^2\theta \:=\:108 \quad\Rightarrow\quad r^2\theta \:=\:216 .[2]


    Divide [2] by [1]: . \frac{r^2\theta}{r\theta} \:=\:\frac{216}{12} \quad\Rightarrow\quad\boxed{ r \:=\:18\text{ cm}}

    Substitute into [1]: . 18\theta \:=\:12 \quad\Rightarrow\quad \boxed{\theta \:=\:\frac{2}{3}\text{ radians}}

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