## Car Car = Fuelish Hyperbolae

My blogoverse buddy (B_{v}B^{TM}) Jonathan asked me to contribute to CoM-18, which he is hosting and I am happy to do so. But first a belated shoutout to my B_{v}B, and best reviewer, Dave Marain, who interviews Professor Lynn Steen, a principal architect of the original NCTM Standards and a highly respected voice in reform mathematics.

Don’t expect any research topics here. This is about solving a practical problem in automotive gas economy which involves a pricing anomoly, a Greek mathmatician who may have tutored Alexander the Great, and an 18th century Scottish math professor who *almost* loses his job by taking an unauthorized 2-year sabbatical to tour the Continent. Can you imagine *not* losing your job today?

So what is the problem? I have 2 cars, an older model which uses regular gas, and a slightly newer model which only uses the more expensive high-test gas, but also gets more miles per gallon (mpg). The question is which is more economical to drive? The pricing anomaly I have observed is that no matter the price per gallon (ppg) of gasoline, and the price has fluctuated up and down quite a bit, the difference in price between regular and high-test is almost always a constant, viz., 25 cents. So which car is more economical to drive actually depends on the price of gasoline! Observe.

The price per mile, ppm = price per gallon / miles per gallon. The percent difference in ppm between regular and high-test,

%diff= (ppm/_{regular}ppm) – 1 = (_{high-test}p/m) – 1,where

p=ppg/_{regular}ppgand_{high-test}

m=mpg/_{regular}mpg._{high-test}

If we take

d=ppg–_{high-test}ppg= $0.25 and_{regular}

m= 25 / 27 and

then plot **ppg _{high-test }**against

**%diff**we get:

So if the price of high-test is below about $3.35 the gas guzzler is more economical fuel-wise, and versa vice. But what is this curve? A cubic polynomial trend line fits it almost perfectly. So is it a polynomial? Zooming out gives us a clue.

It is a rectangular hyperbola (first studied by Menaechmus, a student of Plato, about 350 BC) flipped on the X axis and asymptotic to X = 0 and Y = 0.08 = (1 / **m**) -1. You can easily see this by rewriting the equation for **%diff** as

%diff=A/ppg+_{high-test}Bwhere

A= -d/mand

B= (1 /m) – 1

So this leaves the question of why a cubic polynomial fits the data so well. And here is where our Scottish math professor, Colin MacLaurin, comes in. In 1742 he wrote *A Treatise on Fluxions* (pdf), the first systematic application of Newton’s calculus, in which he shows, among other trigonometric marvels, how the equation for a hyperbola could be closely approximated by truncating an infinite polynomial series.

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