Precalculus I Class Notes, 8/25/98

Depth vs. Clock Time Model

We first predict how the depth of water in a uniform cylinder, with water escaping through a hole in the bottom of the cylinder, will change with respect clock time. We then observe depth vs. clock time for a real flow. Making a table of depth vs. clock time, we note that the rate of depth change varies progressively with clock time and changing depth. We graph depth vs. clock time and note how this varying rate determines the shape of the graph. We also calculate the average rate at which depth changes between each pair of successive data points.

When water flows from a uniform cylinder, how does the depth change with respect to time?  We might predict a depth vs. clock time graph.

We can fill a cylinder with water and time the flow to known depths (90, 80, 70, ... cm in this case).

We can graph our results:

We can calculate the rates at which depth changes.  For example, if it took 10 seconds for the depth to change by 20 cm, the rate would obviously be 2 cm / sec.  We note that this commonsense result was arrived at by dividing the depth change by the time required.  In general, this is how we calculate a time rate.

During the first time interval the depth changed by - 10 cm as the time changed from 0 to 11 sec.  So the average rate was (-10 cm) / (11 sec) = -.91 cm / sec.

During the third interval the depth changed by -10 cm in 14 seconds, so the average rate was (-10 cm) / (14 sec) = -.71 cm / sec.

From clock time t = 0 to t = 130 seconds, depth changed from 90 cm to 10 cm.  So there was a -80 cm change in 130 seconds, for an overall average rate of -80 cm / (130 sec) = -.62 cm / sec.