Related Rates

In some problems we have to find the rate of change of a function, y=f(t) for example, with respect to time t, which is . Sometimes, the relation y=f(t) is given or is easily obtained. In this case, calculating  will be very easy. On the other hand, we find some problems in which there are two variables, x and y, related by an equation that does not involve t, but it can be understood that both x and y are functions of t. In this case, it may be possible to find  or . That will be illustrated in the following examples. Note that the main idea is to find a relation between x and y, and then differentiate both sides implicitly with respect to t.

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

Example

Air is blown into a spherical balloon at the rate of 100 cubic centimeters per second. Find the rate of change of its radius when its radius is 10 centimeters.

Solution

Example

A funnel is in the shape of a right circular cone with its vertex pointing downward, its axis being vertical, the cone’s base is of radius 6cm, and the cone’s height is 12cm. A liquid is poured in the funnel at the rate of . In the same time the liquid is leaking from a hole in the vertex at the rate of

Calculate:

1- The rate at which the liquid level is rising when it is 6 cm deep.

2- The rate at which the radius of the liquid surface is increasing at this moment.

3- The time needed to fill the funnel.

Solution