Carbon 14 is a form of carbon which decays exponentially over time. The amount of Carbon 14 contained in a preserved plant is modeled by the equation $$ f(t) = 10\left(\frac{1}{2}\right)^{ct}. $$ Time in this equation is measured in years from the moment when the plant dies (\(t = 0\)) and the amount of Carbon 14 remaining in the preserved plant is measured in micrograms (a microgram is one millionth of a gram). The number \(c\) in the exponential measures the exponential rate of decay of Carbon \(14\).
We are given \(f(t) = 10\left(\frac{1}{2}\right)^{ct}\). When \(t = \frac{1}{c}\) we find
Commentary
This task is a refinement of ``Carbon \(14\) dating'' which focuses on accuracy. Because radioactive decay is an atomic process modeled by the laws of quantum mechanics, it is not possible to know with certainty when half of a given quantity of Carbon \(14\) atoms will decay. The range of years \(5730 \pm 40\) gives a certain probability (about \(68\) percent) that half of the Carbon \(14\) will decay during this span of years: it is of course possible that the actual half life could be shorter or longer. Each given sample of Carbon \(14\) would have to be treated individually on an experimental basis and if many experiments were conducted, an expected \(68\) percent would give a half-life measured between \(5690\) and \(5770\) years.
While the mathematical part of this task is suitable for assessment, the context makes it more appropriate for instructional purposes. This type of question is very important in science and it also provides an opportunity to study the very subtle question of how errors behave when applying a function: in some cases the errors can be magnified while in others they are lessened.