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Math Help - related rates

  1. #1
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    related rates

    At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 21 knots and ship B is sailing north at 18 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

    I took the derivitive of both sides dx/dt = 21 and dy/dt =18
    and at 4pm, x= 20 +21*4, y=18*4
    21+21+21+21+20 =104
    18*4= 72
    D = sqrt(x^2+y^2)
    D' = 1/2(x^2+Y^2)^-1/2 *(2xx' +2xY')
    Z= sqrt((104+72) + 20^2) = sqrt(31376)

    104+72/sqrt(31376)*(21+18)
    would you let me know if I am on the right path.
    Thankx
    keith
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  2. #2
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    Hello, Keith!

    At noon, ship A is 20 nautical miles due west of ship B.
    Ship A is sailing west at 21 knots and ship B is sailing north at 18 knots.
    How fast (in knots) is the distance between the ships changing at 4 PM?
    (Note: 1 knot is a speed of 1 nautical mile per hour.)

    I have an approach that eliminates dealing with the two different speeds.

    I'll refer to "miles" (nautical miles) and "miles per hour" (knots).
    Code:
                                      * B
                                  *   |
                              *       |
                      x   *           | 18t
                      *               |
                  *                   |
              *                       |
          * - - - - - - - * - - - - - *
          A     21t       P     20    Q

    Ship A starts at point P, 20 miles west of point Q and sails west at 21 mph.
    . . In t hours, it has sailed 21t miles to point A.

    Ship B starts at point Q and sails north at 18 mph.
    . . In t hours, it has sailed 18t miles to point B.

    Let x = distance AB.
    . . . . . . . . . . _________________
    Then: .x .= .√(21t + 20) + (18t) .= .(765t + 840t + 400)^



    Differentiate with respect to time:
    . . dx
    . . --- . = .(765t + 840t + 400)^
    (-)(1530t + 840)
    . . dt


    At 4 PM (t = 4), we have:

    . . dx . . . . . . . 15304 + 840 . . . . . . . . . . . 6960
    . . --- . = . ------------------------------- . = . -------------
    . . dt . . . . 2√7654 + 8404 + 400 . . . . .2√16,000


    Therefore: . dx/dt . .27.5 mph (knots).

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