calculus

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I need help with this problem. I think I figured out how to set it up but I keep getting stuck in the middle.

A boat is pulled into a dock by a rope attatched to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat. If the rope is pulled in at a rate of 1m/s, how fast is the boat approaching the dock when it is 8m from the dock?

You would set up a right triangle and use the pythagoreom therom to solve I think. I need to figure out dx/dt right? How would I do that?

  • calculus -

    yes, label your right-angled triangle this way:
    hypotenuse = h , (the rope)
    vertical = 1 , (height of dock above bow)
    horizontal = x , (path of the boat)

    you are given dh/dt = 1 m/s
    when x = 8, h^2 = 1^2 + 8^
    h = √65

    from x^2 + 1^2 = h^2
    2x dx/dt = 2h dh/dt
    then dx/dt = 2(8)(1)/(2√65)
    = 8/√65 m/s or appr. 0.992 m/s

  • oops typocalculus -

    got my substitution backwards in the last two lines
    should be:

    dx/dt = 2√65(1)/16
    = √65/8 or 1.0079 m/s

  • calculus -

    thanx, that is what i thought i just had trouble sustituting all the numbers back in.

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