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Related Rates and plane

  1. Apr 1, 2007 #1
    1. The problem statement, all variables and given/known data
    This last related rates HW problem is givin me trouble for some odd reason.
    A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 11 km and climbs at an angle of 25 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 1 minutes later?

    2. Relevant equations
    Law of Cosines:
    [tex]c^2=a^2+b^2-2abCos(\theta)[/tex]
    [tex]a=11km[/tex]
    [tex]b=4km[/tex]
    [tex]\frac{da}{dt}=0[/tex]
    [tex]\frac{db}{dt}=4km/min[/tex]

    3. The attempt at a solution
    First using the law of cosines I found c at that particular moment.
    [tex]c=\sqrt(137-88Cos(23\pi/36))[/tex]
    Second I found the derivitive of the law of cosines
    Working everything out I get:
    [tex]\frac{dc}{dt}=\frac{16-44cos(23\pi/36)+44sin(23\pi/36)}{c}[/tex]
    I plug in c and get the wrong answer, what did I do wrong?
     
    Last edited: Apr 1, 2007
  2. jcsd
  3. Apr 2, 2007 #2
    c by the way is the hypotnuse, a in the altitude, and b is length the plane travels,
     
  4. Apr 2, 2007 #3

    HallsofIvy

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    b is NOT "4km". b is a variable and you are told that db/dt= 4 km/min

    Also, there is no reason to convert 25 degrees to [itex]23\pi/36[/itex] since it is a constant. That doesn't change the result but I thought it was peculiar to convert from degrees to radians (and surprised that it was such a simple result!).
     
    Last edited: Apr 2, 2007
  5. Apr 2, 2007 #4
     
  6. Apr 2, 2007 #5

    HallsofIvy

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    Oops! Yes, I skipped over the "1 minute later" part.

    However the point is that is not a "constant"- b is changing as time goes on. You cannot evaluate at b= 4 until after you take the derivative.
    And how did you get that "[itex]sin(23\pi/36)[/itex]"? You don't differentiate the cosine- its a constant.

    The law of cosines tell you that
    [tex]c^2= 11^2+ b^2- 22b cos(115)[/tex]
    Differentiating that with respect to t gives you
    [tex]2c dc/dt= 2b db/dt- 22cos(115) db/dt[/itex]

    Now use the fact that, at this instant, b= 4 km, db/dt= 4 km/min. You will need to determine c, at this instant, from the law of cosines.
     
  7. Apr 2, 2007 #6
    ah! thanks!
     
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