The Lower Mississippi River meanders over its flat valley, forming broad loops called ox-bows. In a flood, the river can jump its banks and cut off one of these loops, getting shorter in the process. In his book Life on the Mississippi (1884), Mark Twain suggests, with tongue in cheek, that some day the river might even vanish! Here is a passage that shows us some of the pitfalls in using rates to predict the future and the past.
In the space of one hundred and seventy six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over a mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oölitic Silurian Period, just a million years ago next November, the Lower Mississippi was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-pole. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo [Illinois] and New Orleans will have joined their streets together and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
Let L be the length of the Lower Mississippi River. Then L is a variable quantity we shall analyze.
L is changing, in miles per year? What approximation does he
use for this rate? Is this a reasonable approximation? Is this rate
positive or negative? Explain. In what follows, use
Twain's approximation.
miles long. How long must the river have been when
he wrote the book?
t is the number of years since 1884. Write a formula
that describes how much L has changed in t years. Your formula
should complete the equation
t
years after 1884. Now write a formula that describes how long the river
is t years later.
L was a million years ago. Does
your answer confirm Twain's assertion that the river was ``upwards of
1,300,000 miles long'' then?
miles long? (This is called a reality check.) What, if
anything, is wrong with the ``trifling investment of fact'' which led to
such ``wholesale returns of conjecture'' that Twain has given us?