Task
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 $14$.

How many micrograms of Carbon $14$ are in the plant at the time it died?

The best known estimate for the halflife of Carbon 14, that is the amount of time it takes for half of the
Carbon $14$ to decay, is $5730 \pm 40$ years. Use this information to
calculate the range of possible values for the constant $c$ in the equation for $f$.

Use your answer from part (b) to find the range of years when there is
one microgram remaining in the preserved plant.
IM 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 halflife 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.