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

Question

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 (in m/s) is the boat approaching the dock when it is 6 m from the dock?

Step 1

Using the diagram below, find the relation between x and y.

Step 2

We must find dx/dt. Differentiating both sides gives us the following

? * dy/dt * = 2x * dx/dt

Step 3

substituting for dy/dt gives the following

2y * (?) = 2x *dx/dt

dx/dt=?

Answer 1

Hey Ashley,

I'm taking a guess as to what the diagram that's mentioned looks like, but this is a pretty standard related rates problem.

First we'd draw a picture. It sounds like your diagram would be a right triangle where the bottom leg is the distance from the dock to the boat along the water, the upright leg would be the fixed 1 meter above the boat to the pulley, and then the hypotenuse would represent the length of rope attaching the boat to the pulley.

Based on context, my guess is that the bottom leg is what we'll call x, and the hypotenuse we'll call y.

In any related rates problem, once we have a good diagram, the next step is to figure out some kind of formula that relates all of the variables.

In our case, we're working with a right triangle, so we can use the good ole Pythagorean Theorem.

With the given info, we'd use:

x² + 1² = y².

Once you come up with your formula, we need to use implicit differentiation, here we're doing it with respect to time (t, our independent variable).

So we take d/dt( x² + 1 = y²).

Go ahead and take that derivative, and you'll end up with your solution to step 2.

Finally, we utilize given information to plug into our answer from step 2. They told us in the question that the rope is being pulled in at 1 m/s (dy/dt). They also told us the boat is 6 m away (that's our x). So in order to solve for what they want (dx/dt), we'll also need to figure out y.

Try drawing out the picture in this scenario to see if you can find y, then plug in x, y, and dy/dt into your equation to solve for dx/dt.

I hope that helps you get started! Feel free to message if you're still having trouble.