## 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?

Function Grapher

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.18 Case Study: Toricelli's Law (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### Interesting Application

The lowest stream has the greatest velocity.

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel