Carbon dating exponential functions
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 \(\Page Index\).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.
Expressed in scientific notation, this is \(4.01134972 × 1013\).To describe these numbers, we often use orders of magnitude.The order of magnitude is the power of ten, when the number is expressed in scientific notation, with one digit to the left of the decimal.In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.
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.
To find \(A_0\) we use the fact that \(A_0\) is the amount at time zero, so \(A_0=10\).