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Math Help - Related Rates

  1. #1
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    Related Rates

    A kite 118 ft above the ground moves horizontally at a speed of 6ft/s. At what rate is the angle between the string and the horizontal decreasing when 236 ft of string have been let out?
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  2. #2
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    We want \frac{d{\theta}}{dt} when dx/dt=6.

    The kite's string makes the hypoteneuse of the triangle. When 236 of string

    is out then the angle is

    sin(\frac{118}{236})=sin(\frac{1}{2})=\frac{\pi}{6  }

    at that instant.

    x=118cot({\theta})

    \frac{dx}{dt}=-118csc^{2}({\theta})\frac{d{\theta}}{dt}

    \frac{d{\theta}}{dt}=\frac{-sin^{2}({\theta})}{118}\frac{dx}{dt}

    Now, using {\theta}=\frac{\pi}{6} \;\ and \;\ \frac{dx}{dt}=6

    \frac{d{\theta}}{dt}=\frac{-sin^{2}(\frac{\pi}{6})}{118}(6)=\frac{-3}{236} \;\ rad/sec
    Last edited by galactus; November 24th 2008 at 05:39 AM.
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  3. #3
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    Hello, asnxbbyx113!

    A kite 118 ft above the ground moves horizontally at a speed of 6 ft/s.
    At what rate is the angle between the string and the horizontal decreasing
    when 236 ft of string have been let out?
    Code:
                       x
          * - - - - - - - - - - - *
          |                 θ  *
          |                 *
          |              *
      118 |           *
          |        *
          |     *
          |  *
        - * - - - - - - - - - - - - -

    We have: . \tan\theta\:=\:\frac{118}{x} \:=\:118x^{-1}

    Differentiate with respect to time: . \sec^2\theta\left(\frac{d\theta}{dt}\right) \:=\:-118x^{-2}\left(\frac{dx}{dt}\right)

    . . And we have: . \frac{d\theta}{dt}\;=\;-\frac{118\cos^2\theta}{x^2}\left(\frac{dx}{dt}\rig  ht) [1]


    When the string is 236 feet, Pythagorus gives us:
    . . x^2 + 118^2 \:=\:236^2\quad\Rightarrow\quad x^2\:=\:41,772\quad\Rightarrow\quad x \:=\:\sqrt{41,772}

    . . Then: . \cos\theta \:=\:\frac{\sqrt{41,772}}{236}\quad\Rightarrow\qua  d\cos^2\theta \:=\:\frac{41,772}{55,696}

    . . And we are told that: . \frac{dx}{dt} \,=\,6 ft/sec.


    Substitute into [1]: . \frac{d\theta}{dt} \;= \;-\frac{118(\frac{41,772}{55,696})}{41,772}(6) \;=\;-\frac{3}{236} radians/sec

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