Carbon dating exponential functions
We may use the exponential decay model when we are calculating half-life, or the time it takes for a substance to exponentially decay to half of its original quantity.We use half-life in applications involving radioactive isotopes.To find \(A_0\) we use the fact that \(A_0\) is the amount at time zero, so \(A_0=10\).To find \(k\), use the fact that after one hour \((t=1)\) the population doubles from \(10\) to \(20\).It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air.It is believed to be accurate to within about \(1\%\) error for plants or animals that died within the last \(60,000\) years.In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph.Exponential growth and decay graphs have a distinctive shape, as we can see in Figure \(\Page Index\) and Figure \(\Page Index\).
On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model.
So, we could describe this number as having order of magnitude \(1013\).
When an amount grows at a fixed percent per unit time, the growth is exponential.
We may use the exponential growth function in applications involving doubling time, the time it takes for a quantity to double.
Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time.