A boat is pulled into a dock by means of a rope attached to a pulley on the dock.

Question

A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 9 feet below the level of the pulley. If the rope is pulled through the pulley at a rate of 14 ft/min, at what rate will the boat be approaching the dock when 110 ft of rope is out?

Answer 1

For this problem you would want to draw a diagram. In this case a right triangle with the right angle on the bottom left corner(sea level), the boat on the bottom right, and the pulley on the top left. The hypotenuse connecting the pulley and the boat is the rope and will be denoted r. The distance between the pulley and the ocean is 9 feet (the vertical leg of the triangle). We will call the x the distance between the boat and the bottom of the dock (the horizontal leg of the triangle) . So using the pythagorean theorem, we can write the equation

1. x²+9²=r²
2. we can differentiate both sides with respect to t: 2xdx/dt=2rdr/dt

Now we are tasked with finding dx/dt when r=110ft and dr/dt=14ft/min. So we need to find x by using the first equation:

1. x²+81=110² => x≈109.63
2. 109.63dx/dt=110*14 => dx/dt = 14.04ft/min