# Carbon dating exponential

26-Aug-2016 04:57

A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the $-0.693$ value, but perhaps my answer will help anyway.If we assume Carbon-14 decays continuously, then $$C(t) = C_0e^{-kt},$$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$\frac{1}{2} C_0 = C_0e^{-5700k},$$ which means $$\frac{1}{2} = e^{-5700k},$$ so the value of $C_0$ is irrelevant.Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.Now, take the logarithm of both sides to get $$-0.693 = -5700k,$$ from which we can derive $$k \approx 1.22 \cdot 10^{-4}.$$ So either the answer is that ridiculously big number (9.17e7) or 30,476 years, being calculated with the equation I provided and the first equation in your answer, respectively. The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places.

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12C (a stable isotope), and carbon-14, denoted 14C (a radioactive isotope).The ratio of the amount of 14C to the amount of 12C is essentially constant (approximately 1/10,000).Carbon 14 is a common form of carbon which decays over time.The amount of Carbon 14 contained in a preserved plant is modeled by the equation  f(t) = 10e^{-ct}.

When an organism dies, the amount of 12C present remains unchanged, but the 14C decays at a rate proportional to the amount present with a half-life of approximately 5700 years.This change in the amount of 14C relative to the amount of 12C makes it possible to estimate the time at which the organism lived.