Carbon 14 is a common form of carbon which decays over time. The amount of Carbon 14 contained in a preserved plant is modeled by the equation $$ f(t) = 10e^{-ct}. $$ 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). So when \(t = 0\) the plant contains 10 micrograms of Carbon 14.
In either case, it is more appropriate to report the time since the plant has died as approximately 19,000 years since these measurements are never completely precise.
If we evaluate this expression on a calculator, we get a value of approximately 19,035 years since the plant has died. Note that if the approximate value \(0.000121\) is used in place of \(\frac{\ln{2}}{5730}\) then an approximate value of 19,030 years is found instead.
In either case, it is more appropriate to report the time since the plant has died as approximately 19,000 years since these measurements are never completely precise.
Commentary
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of \(e\), as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.