In 1966, a Miami boy smuggled three Giant African Land Snails into the country. His grandmother eventually released them into the garden, and in seven years there were approximately 18,000 of them. The snails are very destructive and had to be eradicated. According to the USDA, it took 10 years and cost $1 million to eradicate them.
Since it is assumed that the growth is exponential, we write \(P(t)=ae^{rt}\) for some constants \(a\) and \(r\). We are given two data points, namely that \(P(0)=3\) and \(P(7) =18,000\). (This last value is only an approximation.) From the first data point we get \(3 = P(0) =ae^{0t}=a\), so we must have \(a=3\). Then \(P(7)=18,000\) gives $$ 18,000=3e^{r(7)}. $$ Dividing both sides by 3 and taking the natural logarithm of both sides gives \(7r=\ln(6000)\), so \(r=\frac{\ln(6000)}{7}\approx 1.24\). So \(P(t)=3e^{1.24t}\) for \(0 \le t \le 7\). We use this function to model the population, bearing in mind that it is based on approximate data.
Commentary
The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part. Although the task doesn't point this out explicitly, the teacher can bring it out in discussion, and it illustrates a key property of exponential growth.
Students might choose to model exponential growth using either \(P(t)=ae^{rt}\) or \(P(t)=ab^t\). Each has its advantages and disadvantages. The base \(b\) is the factor by which the population increases each year, and so is useful in part (c). And this interpretation of \(b\) yields directly the fact that \(3b^7 = 18,\!000\), so that it can be calculated without logarithms. On the other hand, students might be more confortable working with base \(e\) and fitting \(P(t) = a e^{rt}\) to two data points, without engaging initially in the reasoning above. Depending on the goal that a teacher has in mind when using this task, he or she might suggest using one form or the other, or might use the opportunity to compare results when different students use different methods.