2.18 Case Study: Torricelli’s Law



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


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?


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Interesting Application

The lowest stream has the greatest velocity.

2.17 Related Rates Table of Contents 3.1 Modeling with Differential Equations

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