2.18 Case Study: Torricelli’s Law

Summary

The purpose of this Case Study in Calculus is to determine how long it would take a cylindrical tank of given dimensions to empty its liquid contents through a bottom outlet hole. Like all CSCs, this represents a real application of calculus. The solution of this practical problem in fact can be traced back to a principle stated by Evangelista Torricelli (1608-1647), a mathematician and physicist who served as Galileo's secretary. In the CSC, calculus is used to model and solve the problem.

By the time you complete the CSC, you should know

The question that needs to be answered.
The relevant information that is available.
How to set up the model and derive the differential equation that will represent the model.
How to solve the differential equation.
How to use the mathematical solution of the differential equation to answer the original question.
How to work with one or more other students on a team project.
How to write a CSC report, especially the Interpretation and Summary.
This CSC as yet another example of applying mathematics you have learned to solving a real-world problem.

Examples

For a given quantity of gas, Boyle's Law states that the pressure exerted by the gas and the volume it takes up are inversely proportional to each other; that is, P = C/V; or equivalently, PV = C. Assume that the volume changes over time, so V is a function of t. Find the rate of change of the pressure with respect to time in two different ways: in terms of V, dV/dt only, and in terms of P, dV/dt only.

Assume that the pressure exerted by the gas is modeled by the equation

Determine the rates at which the pressure and the volume are changing, with respect to time.

A gas occupies 6.52 liters at a pressure of 0.92 atmospheres. Determine the volume if the pressure is increased to 1.44 atmospheres. If the pressure change is linear and occurs over a period of 5 seconds, what is the instantaneous rate of change of the volume with respect to time at 2 seconds?