A dock is 2 meters above water

Question

A dock is 2 meters above water. Suppose you stand on the edge of the dock and pull a rope attached to a boat at the constant rate of a 1 m/s.Assume theboat remains at water level. At whatspeed is the boat approaching the dock when it is 5 meters from the dock? 10 meters from the dock? Isn't it surprising that the boat's speed is not constant?

At 5 meters x′ = ( ); at 10 meters x′ =(  )

Answer 1

Hi Alzaim,

This related rates problem can be solved by setting up a triangle with height 2 m, distance from the dock to the boat, along the water, as x, and distance from the dock to the boat, along the rope, as L, so that L^2=2^2+x^2.

Note that the rate at which the rope is being reeled in, 1 m/s, is dL/dt. However, keep in mind that dL/dt = - 1 m/s, as L is decreasing.

With this information, you can use implicit differentiation to find dx/dt when x=5 m, and then 10 m. dx/dt should also come out negative, but the wording of the problem makes it seem as if it just wants the speed of the boat, which would be positive.

Hope this helps,

John