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Crossing the Street ("dimensionless" equations and Poisson statistics)

I've been going through a new Calculus text and looking over the exercises. This book is consistently poor in the way they handle units of measurement, and they make equations dimensionless at every opportunity. Very sad. I think it does the students a disservice—especially since this is supposed to be Calculus with Applications. It is an excellent opportunity for them to learn how to manage units in real problems.

The exercise below stood out so much that I had to reverse engineer it. That process turned out to be revealing and fun. It certainly reinforced my sense that students are being deprived of a proper treatment of the material, when the authors (or instructors) go to great lengths to sweep the units of measurement under the carpet.

This premise of this exercise is reproduced directly from the text.1 It is quoted for purposes of literary criticism:

Consider a child waiting at a street corner for a gap in traffic that is large enough so that he can safely cross the street. A mathematical model for the traffic shows that if the expected waiting time for the child is to be at most 1 minute, then the maximum traffic flow, in cars per hour is given by

f(x)  =   29,000(2.322 – log x

where x is the width of the street in feet.*

*Bender, Edward, An Introduction to Mathematical Modeling, John Wiley & Sons, 1978, p. 213.

The topic for this month is to reverse-engineer this problem. Except in the blockquote above, I will use x to represent the width of the street, and I will use X to represent the dimensionless quantity x/(1 foot), which appears as x in the textbook exercise. I will also use τ to represent the maximum expected waiting time, which the exercise sets at one minute.

The first step is to express the problem in its natural form, with no "unexplained" numerical constants. (You may find it helpful to recall that "log" normally implies "log10" in math textbooks.) This step requires only an understanding of logarithms.

The second step is to try to derive the formula to infer what assumptions were made. Presumably the assumption is that the traffic follows a Poisson distribution. Anticipating that, let's agree that the traffic flow is represented by λ. That is, "f(x)" in the textbook exercise is related to λ by the expression λ = f(x)/(1 hour). If you make the same derivation that I did, then you will need to use integration and an understanding of the Poisson distribution.

I chose this topic for two reasons. First, the "natural" form of this equation is very simple and elegant (unlike the dimensionless form presented in the textbook exercise). Second, the expression that I got for λ was not solvable analytically, but there is an interesting approximation that can be made to get the simple and elegant equation. The validity of that approximation is interesting.

I haven't had an opportunity to look at the referenced material in Bender's book. It's out of print from the original publisher and I have no idea whether the new version from Dover Publications contains the same material. I don't have any reason to expect that Bender sweeps the units of measurement under the carpet. As far as I know, that was done during the preparation of the calculus textbook.

1Margaret L. Lial, Raymond N. Greenwell, and Nathan P. Ritchey, Calculus with applications: brief version—7th ed., Addison Wesley, 2002, p. 247.

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