In order to use Carbon \(14\) for dating, scientists measure the ratio of Carbon \(14\) to Carbon \(12\) in the artifact or remains to be dated. When an organism dies, it ceases to absorb Carbon \(14\) from the atmosphere and the Carbon \(14\) within the organism decays exponentially, becoming Nitrogen \(14\), with a half-life of approximately \(5730\) years. Carbon \(12\), however, is stable and so does not decay over time.
Scientists estimate that the ratio of Carbon \(14\) to Carbon \(12\) today is approximately \(1\) to \(1,000,000,000,000\).
Suppose \(d\) represents the amount of Carbon \(14\) in the plant at its time of death while \(b\) represents the total amount of Carbon \(12\). As time progresses, \(b\) does not change while \(d\) decreases exponentially. Hence the ratio \((d:b)\) also decreases exponentially. At the time of the plants' death, we have, assuming the same ratio of Carbon \(14\) to Carbon \(12\) as today, that $$ \frac{d}{b} = \frac{1}{1,000,000,000,000}. $$ Let \(f\) be the function which assigns to \(t\), the number of years since the plant's death, the ratio of Carbon \(14\) remaining in the preserved plant to Carbon 12. So \(f(0) = \frac{d}{b} = \frac{1}{1,000,000,000}\). Since the ratio of Carbon \(14\) to Carbon \(12\) is cut in half every \(5730\) years, this means that the equation $$ f(t) = \left(\frac{1}{1,000,000,000}\right) \left(\frac{1}{2}\right)^{\frac{t}{5730}}. $$ models the ratio of Carbon \(14\) to Carbon \(12\) if the input \(t\) is in years and the output \(f(t)\) is measured in micrograms. Indeed, \(f\) satisfies \(f(0) = \frac{1}{1,000,000,000}\) and the values of \(f\) for positive \(t\) decrease exponentially, being cut in half every \(5730\) years.
Commentary
This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon \(14\) isotope remaining in the sample but rather on the ratio of Carbon \(14\) to Carbon \(12\). This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon \(14\), that is, as soon as it dies.
Carbon \(14\) dating is a fascinating topic and much information can be found on Wikipedia.
Many factors limit the accuracy of using Carbon 14 for dating including
This problem is intended for instructional purposes only. It provides an interesting and important example of mathematical modeling with an exponential function. If the teacher has the time and inclination, it also reveals many of the inherent difficulties with mathematical modeling, some of which are mentioned in the previous paragraph as regards this particular example.