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15 January, 18:21

At noon, ship a is 180 km west of ship

b. ship a is sailing east at 35 km/h and ship b is sailing north at 30 km/h. how fast is the distance between the ships changing at 4:00 pm?

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  1. 15 January, 18:25
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    If we let the coordinates of each ship be (x, y) with the positive directions of these coordinates corresponding to East and North, then the positions of the ships t hours after noon are

    Ship A

    (-180+35t, 0)

    Ship B

    (0, 30t)

    The distance between them is the "Pythagorean sum" of the difference in their coordinates:

    d = √ ((-180 + 35t) ² + (-30t) ²)

    = √ (32400 - 12600t + 2125t²)

    The rate of change of this distance is

    dd/dt = (2125t - 6300) / √ (32400 - 12600t + 2125t²)

    At 4 pm, the value of this rate of change is

    (2125*4 - 6300) / √ (32400 - 12600*4 + 2125*4²)

    = 2200/√16000

    ≈ 17.39 km/h

    The distance between the ships is increasing at about 17.39 km/h at 4 pm.
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