Chapter 2.3, Problem 15P

An important tool in archaeological research is radiocarbon dating, developed by the American chemist Willard F. Libby. This is a means of determining the age of certain wood and plant remains, hence of animal or human bones or artifacts found buried at the same levels. Radiocarbon datingis based on the fact that some wood or plant remains contain residual amounts of carbon- $14$, a radioactive isotope of carbon. This isotope is accumulated during the lifetime of the plant and begins to decay at its death. Since the half-life of carbon- $14$ is long (approximately $5,730$ years), measurable amounts of carbon- $14$ remain after many thousands ofyears. If even a tiny fraction of the original amount of carbon- $14$ is still present, then by appropriate laboratory measurements the proportion of the original amount of carbon- $14$ that remains can be accurately determined. In other words, if $Q\left(t\right)$ is the amount of carbon- $14$ at time $t$ and $Q_0$ is theoriginal amount, then the ratio $Q\left(t\right)/Q_0$ can be determined, at least if this quantity is not too small. Present measurementtechniques permit the use of this method for time periods of $50,000$ years or more.

Assuming that $Q$ satisfies the differential equation $Q\text{'}=-\text{}r\text{}Q$, determine the decay constant $r$ for carbon- $14$.

Findan expression for $Q\left(t\right)$ at any time $t$, if $Q\left(0\right)=Q_0$.

Suppose that certain remains are discovered in which the current residual amount of carbon- $14$ is $50\%$ of the original amount. Determine the age of these remains.