Given a boat sailing at a constant speed of 12 miles per hour. The boat is 4 miles offshore, sailing parallel to a straight coast. How fast is it approaching a lighthouse on the shore when it is exactly 5 miles from the lighthouse?

Let

The following diagram illustrates the setup:

The problem gives us that the distance along the shore to the lighthouse is changing at a rate of 12 miles per hour (since the boat is moving at 12 miles per hour and stays parallel to the straight shoreline). Thus, . Then, using the Pythagorean theorem, when we have . Furthermore, solving in terms of and differentiating we have

So, we then have at ,

Small error. In the third from the last line it should be dx/dy not dy/dx