A spherical container of r meters is being filled with a liquid at a rate of . At what rate is the height of the liquid in the container changing with respect to time?
Let V be the volume already in the container, h be the height of the liquid in the container and t be the time since some initial starting point. We are given that and we are asked for the related rate . The other rate needed for this problem is the rate at which the height is changing with respect to the volume, namely . In order to calculate this, we need a relation between the height and the volume of the liquid at any particular point in time.
We will use a trick from integral calculus to get the volume in terms of the height. Let a silhouette of our container be described by the implicit equation x2 + (y − r)2 = r2, a circle of radius r whose bottom is on the x-axis. Using the slicing method for finding a volume of revolution, we get that . This simplifies into the formula . Then , so .
Then we get the general forumla for the desired rate: . This is a general formula which will give us the rate at which the height is changing at any given height (assumed to be between 0 and 2r).