Calculus

At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h, and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 P.M.?

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  1. mark position A and B, where AB = 150

    At a time of t hrs, place the shipA at a point P, between A and B, and place shipB at Q, where Q is vertically above B
    so , using Using distance = rate x time
    AP = 35t, and BQ = 25t
    PBQ is a right-angled triangle
    then
    PQ^2 = (150 - 35t)^2 + (25t)2
    2 PQ d(PQ)/dt = 2(150-35t)(-35) + 2(25t)(25)
    at 4:00 pm, t = 4
    then PQ^2 = 10^2 + 100^2
    PQ = √10100

    in the derivative equation, when t = 4 , and PQ=√10100
    2√10100 dPQ/dt = 2(10)(-35) + 5000
    dPQ/dt = 4300/2√10100
    = appr 21.4 km/h

    distance is increasing at appr 21.4 km/h

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