Exponential growth and decay (Part 1): Phrasing of homework questions

I just completed a series of posts concerning the different definitions of the number e. As part of this series, we considered the formula for continuous compound interest

A = Pe^{rt}

Indeed, this formula can be applied to other phenomena besides the accumulation of money. Unfortunately, as they appear in Precalculus textbooks, the wording of questions involving exponential growth or decay can be either really awkward or mathematically imprecise (or both). Here’s a sampling of problems that I’ve collected from various sources:

One thousand bacteria on a petri dish are placed in an incubator, encouraging a relative rate of growth of 10% per hour. How many bacteria will there be in two days?

This is mathematically precise, as it relates to the differential equation A'(t) = r A(t) with solution A = P e^{rt}. The meaning of the value of r is clear from dimensional analysis: the units of A'(t) are \hbox{bacteria}/ \hbox{hour}, while the units of A(t) are \hbox{bacteria}. Therefore, the units of r must be \hbox{hour}^{-1}. So saying that there’s a “relative rate of growth of 10% per hour” makes total sense.

Of course, when Precalculus students are solving this problem, they have no idea about what a differential equation is, making the word relative seem superfluous to the problem.

A sum of $5000 is invested at an interest rate of 9% per year. Find the time required for the money to double if the interest is compounded continuously.

What the problem is trying to say is “Let r = 0.09.” But this is a horrible way to write this in ordinary English! After all, if we plug r = 0.09 and t = 1 into the formula, we obtain

A = P e^{0.09 \times 1} \approx 1.09417P

So it would appear that the interest rate after one year is about 9.417%, and not 9%.

Indeed, if we read the problem at face value that the interest rate is 9% per year, then it stands to reason that, after one year, we have

P(1.09) = P e^{r \cdot 1}

1.09 = e^r

\ln 1.09 = r

In a nutshell, saying that there is “an interest rate of 9% per year” can easily be interpreted to mean that the annual percentage rate is 9% year, and this can be a conceptual barrier for literally-minded students.

I don’t have a good solution for this impasse between ordinary English and giving clear directions to students about what numbers should be used in the formula. But I do think that it’s important for teachers to be aware of this possible misunderstanding as students read their homework questions.

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