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Math Help - Area and limit of the shaded region...?

  1. #1
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    Area and limit of the shaded region...?

    I'm not sure if I did this is right, but here's the question:

    Consider the shaded region outside the sector of a circle of radius 10 meters and inside a right triangle.

    a) write the area A= f(θ) of the region as a function of θ.

    My equation: f(θ) = 50θ - 50 arctanθ

    b) What is the domain of the function?

    All real numbers.

    c) Find lim (θ--> π/2-) A.

    Infinity... why?

    Thanks!
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  2. #2
    MHF Contributor

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    Quote Originally Posted by damselfly View Post
    a) write the area A= f(θ) of the region as a function of θ.

    My equation: f(θ) = 50θ - 50 arctanθ
    Not quite. The area of the triangle is 50\tan\theta, from which you substract the area of the piece of disc ( 50\theta, indeed), so that A=50(\tan\theta-\theta).

    Now you can reconsider the other questions.

    Laurent.
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  3. #3
    Super Member

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    Hello, damselfly!


    Consider the shaded region outside the sector of a circle of radius 10 meters
    and inside a right triangle.
    Code:
        B *
          |:*  D
          |:::*
        h |:*   *
          |*      *
          |      θ  *
        C * - - - - - * A
               10

    a) Write the area A of the region as a function of \theta.

    In right triangle BCA\!:\;\;\tan\theta = \frac{h}{10} \quad\Rightarrow\quad h \:=\:10\tan\theta

    Area of \Delta BCA \:=\:\frac{1}{2}(10)(10\tan\theta) \:=\:50\tan\theta

    Area of sector ACD \;=\;\frac{1}{2}(10^2)\theta \;=\;50\,\theta

    Therefore: . A \;=\;50\tan\theta - 50\,\theta \quad\Rightarrow\quad\boxed{ A \;=\;50(\tan\theta - \theta)}



    b) What is the domain of the function?

    \theta could be any acute angle from 0 to less than 90.

    . . .  0 \:\leq \:\theta \:<\:\frac{\pi}{2}



    c) Find: . \lim_{\theta\to\frac{\pi}{2}^-} A

    Answer: Infinity . . . why?

    As \theta increases to \frac{\pi}{2}, the radius AD becomes vertical.
    Code:
          |:::::::::|
          |:::::::::|
          |:::::::::|
          |:::::::* * D
          |:::*     |
          |:*       |
          |*        |
          |       θ |
        C * - - - - * A
               10

    And the area of the "triangle" becomes infinite.

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  4. #4
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    Red face

    Thank you so much!
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