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Math Help - Studying for final and need help with Related Rates

  1. #1
    Member FalconPUNCH!'s Avatar
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    Studying for final and need help with Related Rates

    I hate word problems and I pretty much suck when it comes to them. Here's one that I got stuck at.

    1) A man starts walking north at 4ft/s from a point P. Five minutes later a woman starts walking south at 5ft/s from a point 500ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking?

    I don't even know where to begin. My final will be filled with these types of question any tips on how to approach these types of problems?
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  2. #2
    GAMMA Mathematics
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    Tip... draw out what is given in the problem. In related rates & optimization problems, you tend to work with one or two formulas and take a derivative of one of them.
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  3. #3
    Member FalconPUNCH!'s Avatar
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    Quote Originally Posted by colby2152 View Post
    Tip... draw out what is given in the problem. In related rates & optimization problems, you tend to work with one or two formulas and take a derivative of one of them.
    I usually do that, and whenever I can list what x =, y=, and the rates of change, but sometimes I don't know what goes with what.
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  4. #4
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    Hello, FalconPUNCH!

    Make a sketch!


    1) A man starts walking north at 4ft/s from a point P.
    Five minutes later a woman starts walking south at 5ft/s from a point 500ft due east of P.
    At what rate are the people moving apart 15 min after the woman starts walking?
    Code:
        M *
          | *
       4t |   *
          |     *
        A *       *
          |         *
          |           *
     1200 |             *
          |               *
          |                 *
        P * - - - - - - - - - * - - - * Q
          :                     *     |
       5t :                       *   | 5t
          :                         * |
        R * - - - - - - - - - - - - - * W
                      500

    The man starts at P and has a 5-minute headstart.
    In 5 minutes (300 sec), he walks 1200 feet to point A.
    In the next t seconds, he walks 4t ft to point M.

    The woman starts at Q and walks south at 5 ft/s.
    In the next t seconds, she walks 5t ft to point W.

    Let x \,=\,MW
    In right triangle MRW, we have: . MW^2 \:=\:MR^2 + RW^2
    . . That is: . 4t+1200 + 5t)^2 + 500^2 \:=\9t+1200)^2 + 500^2" alt="x^2\:=\4t+1200 + 5t)^2 + 500^2 \:=\9t+1200)^2 + 500^2" />

    Differentiate with respect to time: . 2x\left(\frac{dx}{dt}\right) \:=\:2(9t + 1200)

    . . Hence: . \frac{dx}{dt} \;=\;\frac{9t + 1200}{x}


    In 15 minutes, t = 900
    Find x at that time, and you're done!

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  5. #5
    Member FalconPUNCH!'s Avatar
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    Quote Originally Posted by Soroban View Post
    Hello, FalconPUNCH!

    Make a sketch!


    Code:
        M *
          | *
       4t |   *
          |     *
        A *       *
          |         *
          |           *
     1200 |             *
          |               *
          |                 *
        P * - - - - - - - - - * - - - * Q
          :                     *     |
       5t :                       *   | 5t
          :                         * |
        R * - - - - - - - - - - - - - * W
                      500

    The man starts at P and has a 5-minute headstart.
    In 5 minutes (300 sec), he walks 1200 feet to point A.
    In the next t seconds, he walks 4t ft to point M.

    The woman starts at Q and walks south at 5 ft/s.
    In the next t seconds, she walks 5t ft to point W.

    Let x \,=\,MW
    In right triangle MRW, we have: . MW^2 \:=\:MR^2 + RW^2
    . . That is: . 4t+1200 + 5t)^2 + 500^2 \:=\9t+1200)^2 + 500^2" alt="x^2\:=\4t+1200 + 5t)^2 + 500^2 \:=\9t+1200)^2 + 500^2" />

    Differentiate with respect to time: . 2x\left(\frac{dx}{dt}\right) \:=\:2(9t + 1200)

    . . Hence: . \frac{dx}{dt} \;=\;\frac{9t + 1200}{x}


    In 15 minutes, t = 900
    Find x at that time, and you're done!

    LOL my sketch didn't look like that. But I understand what to do. Usually most have something to do with triangles, so I'm expecting my professor to do one with triangles. Thanks for helping me out with the example.
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