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March 29, 2017

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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 1 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?

  • Calculus - ,

    If the rope is of length h, and the boat is x away from the dock,

    h^2 = 1+x^2
    When x=8, h = √65

    2h dh/dt = 3x dx/dt
    Since dh/dt = -1,

    2√65 (-1) = 3*8 dx/dt
    dx/dt = -√65/12

    Since dx/dt < 0, the boat is approaching the dock at about 0.67 m/s

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