Task
A preserved plant is estimated to contain $1$ microgram (a millionth of a gram) of
Carbon $14$. The amount of Carbon 14 present
in the preserved plant is modeled by the equation
$$
f(t) = A\left(\frac{1}{2}\right)^{\frac{t}{5730}}
$$
where $t$ denotes time since the death of the plant, measured in years,
and $A$ is the amount of Carbon $14$ present in the plant at death, measured
in micrograms.

How much Carbon $14$ was present in the living plant assuming it died $5000$ years ago?

How much Carbon $14$ was present in the living plant assuming it
died $10000$ years ago?

The halflife of Carbon $14$ is the amount of time it takes for half of the
Carbon $14$ to decay. What halflife does the expression for the function $f$
imply for Carbon $14$?
IM Commentary
In the task ''Carbon $14$ Dating'' the amount of Carbon $14$ in a preserved
plant is studied as time passes after the plant has died. In practice,
however, scientists wish to determine when the plant died and, as this
task shows, this is not
possible with a simple measurement of the amount of Carbon $14$ remaining in the preserved plant. Carbon $14$ dating requires many other hypotheses as will be
addressed in ''Carbon $14$ dating in practice II.''
The equation for the amount of Carbon $14$ remaining in the preserved plant
is in many ways simpler here, using $\frac{1}{2}$ as a base. This base is particularly convenient because the exponential rate of decay is determined
by the halflife which, in this case, is seen in the denominator of the exponent,
as shown in part (c) of the task.
This should be contrasted with the equation in the task ''Carbon $14$ dating.''
All three parts of this task can be used for assessment or for instruction although
the intention of this task is as a prelude to ''Carbon $14$ dating in practice II''
where the actual method used by scientists is discussed.
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