how fast is the boat approaching he dock when it is 3 meters away from the dock?

Question

A boat is pulled into a dock by rope attached to it and passing through a pulley on the dock positioned 4 meters higher than the boat. If the rope is being pulled in at a rate of 3m/sec, how fast is the boat approaching he dock when it is 3 meters away from the dock?

Answer 1

At the instant in question, the rope is the hypotenuse "r" of a 3-4-5 triangle, so there are 5 feet of rope between the boat and the pulley. The horizontal distance "x" covered by the rope is changing, but the vertical distance "y" is not.

So we have:

x² + y² = r²

Take d/dt of both sides, remembering that y is constant.

2x*dx/dt + 0 = 2r*dr/dt

x = 3 m, r = 5 m, and dr/dt = -3 m/s (negative because the length of the rope is decreasing). dx/dt is the velocity with which the boat is approaching the dock, and that's what you were asked for.

dx/dt = r/x * dr/dt = (5 m/3 m)*(-3 m/s) = -5 m/s

So the boat is approaching the dock at 5 m/s.

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