Example

A man is standing on the top of a cliff near the sea, which is 150 meters high. He sees a boat, which is 360 meters away from the base of the cliff the boat was moving towards the cliff at 5 meters per second. Find the rate at which the distance between the boat and the observer changes. Assume that the man is standing vertically over the base of the cliff.

Solution

In the given drawing, B is the base of the cliff, A is the point where the observer stands, and C is the position of the boat. We have used x for the distance between the boat and the base of the cliff and y for the distance between the boat and the observer. We can see that triangle ABC has a right angle at B. So, we can write the following relation:

By differentiating both sides of the equation implicitly with respect to t and dividing both sides by 2:

Now, when we consider the given case when  (Notice that the boat is getting closer, so the rate is negative.) and x=360 m. By substituting in the first equation, we can get y=390 m. Now by substituting in the second equation we can get:

This is the rate we want.