The Human Cough 2nd Edition (UMAP)

The human cough contracts the trachea to speed airflow. This Module models the cough as laminar airflow through an elastic pipe, and Hooke’s Law for stretching a spring is applied to contraction of the trachea. Simple differential calculus determines that the average airflow speed is greatest when the trachea contracts to two-thirds of its original radius—a result that agrees with experimental evidence.

EDITOR’S NOTE: This is the 2nd edition of a Module that was originally published as a separate typed fascicule in 1978 and reprinted in 1981 in UMAP Modules 1980: Tools for Teaching, 581–593 (Boston, MA: Birkhauser). This 2nd edition has been typeset without significant changes.

Table of Contents:

1. When You Cough.
2. Notation for a Model of Coughing
3. Laminar Flow
4. Average Velocity and Fluid Flow
5. Perfect Elasticity
6. What Radius R Makes V the Largest?
7. Acknowledgment
8. Solutions to the Exercises 
Reference 
About the Author

Note: The information below was created with the assistance of AI.

Mathematics Level

This module is designed for first-year undergraduate calculus students, especially those learning how to use derivatives to find maximum and minimum values of functions. It may also be appropriate for advanced high school calculus, AP Calculus AB/BC enrichment, or independent reading by stronger students. The mathematics is not highly advanced, but it requires students to connect calculus with a real biological and physical system. The module’s official mathematical field is Calculus.

Application Areas

The module applies mathematics to physics, biology, and medical science. It models the human cough by treating the trachea as an elastic pipe and the airflow as laminar flow. The physical ideas include air pressure, fluid flow, elasticity, Hooke’s Law, and Poiseuille’s Law. The biological and medical context is the contraction of the trachea during coughing, which increases airflow speed and helps clear foreign objects from the airway. The module shows that airflow speed is maximized when the trachea contracts to about two-thirds of its original radius, matching experimental observations.

Prerequisites

Students should know:

• Differentiation of polynomials
• How to interpret $dy/dx=0$
• The second derivative test for maxima and minima
• Operations with inequalities
• Basic curve sketching
• Basic function notation and domain restrictions

Helpful background includes introductory physics concepts such as pressure, force, flow rate, elasticity, and Hooke’s Law, but the module explains the needed context. Integral calculus is mentioned through Poiseuille’s Law, but the main student work uses differential calculus rather than integration.

Subject Matter

The subject matter is an applied calculus model of coughing. The module begins with the biological process of coughing, then introduces variables for the rest radius of the trachea, contracted radius, airflow velocity, pressure difference, and flow rate. It assumes laminar airflow through a circular pipe and perfect elasticity of the tracheal wall. Using these assumptions, the model derives an airflow velocity function of the form

V=c2(R0−R)R2V = c_2(R_0 - R)R^2V=c2​(R0​−R)R2

and asks students to determine which contracted radius $R$ maximizes airflow speed. Students use derivatives, critical points, endpoint testing, and the second derivative test to show that the maximum occurs at R=2R03R = \frac{2R_0}{3}R=32R0​​. The module also includes diagrams of laminar flow, spring elasticity, the tracheal wall model, and the graph of the cubic function used in the optimization.

Correlation to Mathematics Standards

This module aligns strongly with calculus and mathematical modeling standards. For AP Calculus AB/BC, it connects directly to derivatives, optimization, critical points, the second derivative test, interpreting functions in context, and using calculus to solve real-world problems. For Common Core High School Mathematics, it supports modeling, interpreting functions, creating equations, reasoning with quantities, and analyzing polynomial functions. For NCTM Process Standards, it supports problem solving, reasoning, mathematical connections, representation, and communication.

At the undergraduate level, it aligns well with MAA CUPM recommendations because it emphasizes applied calculus, interdisciplinary modeling, quantitative reasoning, and communication of mathematics. It also fits SIAM/COMAP modeling practices by having students define assumptions, identify variables, build a mathematical model, analyze the model with calculus, interpret the result biologically, and compare the conclusion with experimental evidence.

Overall Classification

The Human Cough is best classified as an introductory undergraduate applied calculus module with strong connections to physics, biology, and medical science. Its main educational value is showing students how a simple calculus optimization problem can explain a real physiological behavior: the body contracts the trachea during a cough in a way that maximizes airflow speed.