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

Let r stand for the radius and V stand for the volume of the balloon. The relation between r and V is:

By differentiating both sides with respect to t, we have:

We now have  and . By substituting in the formula above, we can know

.

Note well that we must differentiate the formula we have before substituting by the given values.