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Math Help - Related Rates 4

  1. #1
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    Related Rates 4

    4) You are videotaping a race from a stand 123 feet away from the track, following a car that is moviing at 180 mi/h (264 ft/sec). How fast will your camera angle theta be changing when the car is right in fron of you? A half second later?

    Huge thanks to anyone in advance for any help given
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  2. #2
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    Hello, Super Mallow!

    You are videotaping a race from a stand 123 feet away from the track,
    following a car that is moviing at 180 mi/h (264 ft/sec).
    How fast will your camera angle theta be changing
    (a) when the car is right in fron of you?
    (b) a half second later?
    Code:
          B     x     C
          * → → → → → *
          |         /
          |       /
      123 |     /
          |   /
          |θ/
          * 
          A

    You are at A, 123 feet from the track: AB = 123

    The car is moving to the right from B to C
    . . x = BC .and . \frac{dx}{dt} \,=\,264 ft/sec
    Let \theta \:=\:\angle BAC

    From right triangle ABC, we have: . \tan\theta \:=\:\frac{x}{123}

    Differentiate with respect to time: . \sec^2\!\theta\left(\frac{d\theta}{dt}\right)\:=\:  \frac{1}{123}\left(\frac{dx}{dt}\right)
    . . and we have: . \frac{d\theta}{dt} \:=\:\frac{\cos^2\!\theta}{123}\left(\frac{dx}{dt}  \right) .[1]


    (a) When t=0\!:\;x = 0,\:\theta\,=\,0,\:\cos^2\!0 \,=\,1

    Substitute into [1]: . \frac{d\theta}{dt}\:=\:\frac{1^2}{123}(264) \:\approx\:\boxed{2.15\text{ radians/second}}


    (b) When t = \frac{1}{2}\!:\;x = 132

    Right triangle ABC has sides x = 132 and AB =123.
    . . Its hypotenuse is: . AB \:=\:\sqrt{132^2+123^2} \:=\:\sqrt{32,553} \:\approx\:180.42
    Hence: . \cos\theta \:=\:\frac{123}{180.42} \:=\:0.681742601\quad\Rightarrow\quad \cos^2\!\theta \:\approx\:0.4648

    Substitute into [1]: . \frac{d\theta}{dt}\:=\:\frac{0.4648}{123}(264) \:=\:0.997619512

    Therefore: . \frac{d\theta}{dt}\:\approx\:\boxed{1\text{ radian/second}}

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  3. #3
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    I'm back after 2 days, sorry for not getting to this earlier (Comcast...).

    I get the solution to A. I do not understand B - How do we know X=132 when T=1/2?
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