How so scientist use radioactive decay to date fossils and artifacts

The most important example of radioactive decay used to date fossils and artifacts is the decay of Carbon-14.

Carbon-14 is constantly being created in the atmosphere, and it is constantly taken by all living organisms by heating/breathing. At the same time, carbon-14 also decays, so that in living organisms the amount of carbon-14 remains constant. However, when the organism dies, it does not take anymore carbon-14 from the environment, so the amount of carbon-14 starts to decrease according to a precise rule.

Carbon-14 has a half-life of 5730 years: this means that after this time, the amount of carbon-14 left in the dead organism is half the initial amount. Based on this fact, it is possible to infer the age of a fossil by measuring the amount of carbon-14 left. The relationship between the amount of Carbon-14 left at time t, N(t), and the amount of carbon-14 initially present in the fossil is given by

where  is the half life of carbon-14. By measuring N(t) and by knowing , it is possible to calculate t, the age of the fossil.