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?

Answer 1

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