Carbon dating physics problem
Thus $$ f = \exp[-\lambda \tau]$$ $$ \ln f = -\lambda \tau$$ $$ \frac = |-\lambda \delta\tau |$$ $$ \delta \tau = \frac \frac$$ So say your ability to measure $f$ was limited to $\pm 0.02$ because of potential contamination or other complications, then $$ \delta \tau = \frac\ \tag$$ If $f=0.5$ (i.e something that is just 5730 years old), then your uncertainty would be a perhaps tolerable $\pm 330$ years.
However, if $f=0.0079$ (for 40,000 years old), then the uncertainty would be a less-than-useful $\pm 20,800$ years.
Exponential Decay Formula: A = A" is the original amount of the radioactive isotope that is measured in the same units as "A." The value "t" is the time it takes to reduce the original amount of the isotope to the present amount, and "k" is the half-life of the isotope, measured in the same units as "t." The applet allows you to choose the C-14 to C-12 ratio, then calculates the age of our skull from the formula above.
Archaeologists use the exponential, radioactive decay of carbon 14 to estimate the death dates of organic material.
For example, let's say you can measure the 14/12 C ratio to be $f \pm \delta f$ (in a system of units where the original ratio was expected to be 1).
Crudely speaking, what you do next is to extrapolate a decay curve back in time to see how long ago the sample would have had $f=1$.
At any particular time all living organisms have approximately the same ratio of carbon 12 to carbon 14 in their tissues.
However, given that the half life of carbon 14 is 5730 years, then there really isn't much carbon 14 left in a sample that is 40,000 years old.After the organism dies, carbon-14 continues to decay without being replaced.To measure the amount of radiocarbon left in a artifact, scientists burn a small piece to convert it into carbon dioxide gas.For an example, when they tried to get the carbon dating for presence of Aboriginal people in Australia they get to the number 40,000. Why is that 40,000 years limit for carbon dating methods?Carbon-14 makes up about 1 part per trillion of the carbon atoms around us, and this proportion remains roughly constant due to continual production of carbon-14 from cosmic rays.