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Math Help - Related Rates Problems

  1. #1
    Member FalconPUNCH!'s Avatar
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    Related Rates Problems

    1. A spotlight on the ground shines on the wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

    2. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock.


    So far related rates problems have been pretty easy since we have been using geometric figures. I can't come up with an equation for either of these so I can solve for the change of rate. Can someone help me come up with the equations and show me how to do it in case I get something similar to these in the future?
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by FalconPUNCH! View Post
    1. A spotlight on the ground shines on the wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

    So far related rates problems have been pretty easy since we have been using geometric figures. I can't come up with an equation for either of these so I can solve for the change of rate. Can someone help me come up with the equations and show me how to do it in case I get something similar to these in the future?
    1) using similar triangles, you have
    \frac{x}{2}=\frac{12}{y}
    which implies that
    xy=24
    and
    \frac{dy}{dt} x + \frac{dx}{dt} y =0
    use the given and evaluate at x=12-4=8 (note that if x=8, then y=3)

    for 2) what do you min by "... m higher than..."?
    Attached Thumbnails Attached Thumbnails Related Rates Problems-sketch.jpg  
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  3. #3
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    Hello, FalconPUNCH!

    You left out a measurement in #2.
    . . I'll pick a convenient value . . .


    2. A boat is pulled into a dock by a rope attached to the bow of the boat and
    passing through a pulley on the dock that is 6 m higher than the bow of the boat.
    If the rope is pulled in at a rate of 1 m/s,
    how fast is the boat approaching the dock when it is 8 m from the dock?
    Code:
                                  * P
                              *   |
                     R    *       |
                      *           | 6
                  *               |
              *                   |
        B *   *   *   *   *   *   *
                      x

    The boat is at B, the pulley is at P.

    The length of the rope is: .  R \,=\, BP . and . \frac{dR}{dt} \,=\, -1\text{ m/s}

    From Pythagorus, we have: . x^2 + 6^2 \:=\:R^2

    Differerentiate with respect to time: . 2x\left(\frac{dx}{dt}\right) \:=\:2R\left(\frac{dR}{dt}\right)
    . . and we have: . \frac{dx}{dt} \:=\:\frac{R}{x}\left(\frac{dR}{dt}\right)

    Can you finish it now?

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  4. #4
    Member FalconPUNCH!'s Avatar
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    Oh sorry I left out 1 m. Thanks for your help. I understand now.
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