A spherical container of 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 be the volume already in the container, be the height of the liquid in the container and 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 , a circle of radius whose bottom is on the -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).