### Carbon Dating

Carbon dating is a technique used by scientists to date fossils. It relies on the principle that the C14 isotope of carbon decays (this is called radioactive decay) into another isotope of carbon, C12 at a rate proportional to its mass. This means that it has a constant relative rate of decay. The same equation is used for this situation as for continuous compounding and population growth. With situations involving decay, the rate of growth is always negative.

The half-life of a substance is the amount of time it takes for half of that substance to decay. It is only a property of substances that decay at a rate proportional to their mass. Through research, scientists have agreed that the half-life of C14 is approximately 5700 years. The decay constant, k, for carbon-14 can be calculated using the half-life. Using the constant relative rate of decay function, we say that the remaining amount of carbon-14 C(t) = C(0)e-kt. k is negative because C(t) decreases as t increases.

half-life

 C(5700) = C(0)e-5700k = C(0)

 e-5700k =

 ln e-5700k = ln

 -5700k = ln

 - k =

With a little manipulation, the function C(t) = C(0)e-kt can now be simplified to C(t) = C(0). Then, given the estimated percentage of the original amount of C14 left in an organism, its age can be approximated.