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
![\[ x = \sqrt{y^2 + 16} \quad \implies \quad \frac{dy}{dx} = \frac{y}{\sqrt{y^2 + 16}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8c98ec15b7133f3edfdd89138c7e9df4_l3.png "Rendered by QuickLaTeX.com")
So, we then have at ,
![\[ \frac{dx}{dt} = \frac{dy}{dt} \cdot \frac{dx}{dy} = (12) \left( \frac{3}{\sqrt{9+16}} \right) = \frac{36}{5} = 7.2 \text{ mph}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-10d414a6ac855956950d442fc90b04b7_l3.png "Rendered by QuickLaTeX.com")
Small error. In the third from the last line it should be dx/dy not dy/dx