Task
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
halflife 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 $1,000,000,000,000$.

Assuming that this ratio has remained constant over time,
write an equation for a function which models the ratio of Carbon $14$ to Carbon $12$ in a preserved plant $t$ years after plant has died.

In a particular preserved plant, the ratio of Carbon $14$ to Carbon $12$ is
estimated to be about $1$ to $13,000,000,000$. What can you conclude
about when plant lived? Explain.

Dinosaurs are estimated to have lived from about $230,000,000$ years ago
until about $65,000,000$ years ago. Using this information and the given halflife of Carbon 14, explain why this method of dating is not used for dinosaur remains.
IM 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

the hypothesis that levels of Carbon $14$ in the environment have been relatively constant. These levels can be influenced by climate, by natural processes
such as volcanoes, and in recent times, by human activity.

the accuracy of measurement for the amount of Carbon $14$ in a given sample.
This is a serious issue because the current ratio of $1$ to $1,000,000,000$
means that extremely precise measurements will be needed to determine how much Carbon $14$ is in a specimen.

the method used to estimate the amount of Carbon $14$ in a given sample. If
this is done by measuring the current decaying Carbon $14$ then it is not statistically reliable with very small samples. More recent technology actually
allows scientists to measure the remaining Carbon $14$ much more accurately.
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.