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Related Rate Problem (Involving Trig.)

  1. May 14, 2013 #1
    1. The problem statement, all variables and given/known data

    A rocket is moving into the air with a height function given by h(t) = 200t^2. A camera located 150 m away from the launch site is filming the launch. How fast must the angle of the camera be changing with respect to the horizontal 4 seconds after liftoff?

    2. Relevant equations



    3. The attempt at a solution

    If we create a diagram, we will see that
    tan(θ)=(200t^2)/150 or (4t^2)/3

    Differentiating with respect to t,

    sec^2(θ)dθ/dt=8t/3 which becomes dθ/dt=8t/3 * cos^2(θ)

    At t=4s, tan(θ)=64/3, and then by sinθ=cosθtanθ, we know sinθ=(64/3)cosθ

    Then by sin^2(θ)+cos^2(θ)=1, we know that cos^2(θ)=9/4105

    Now evaluating the derivative at t=4s, we obtain dθ/dt=96/4105 rad/s≈0.0234 rad/s

    I would just like to know if all my steps are accurate, and if my final answer is correct, or if I made an error along the way, leading to an incorrect result?
     
  2. jcsd
  3. May 14, 2013 #2

    LCKurtz

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    Looks OK to me. You could have saved a couple of calculations by using ##\sec^2\theta=1+\tan^2\theta## instead of messing with the sines and cosines.
     
  4. May 15, 2013 #3
    Okay thanks, and I completely forgot about that identity when doing this problem.
     
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