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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 \(\Page Index\).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.The formula is derived as follows \[\begin \dfrac A_0&= A_0e^\ \dfrac&= e^ \qquad \text A_0\ \ln \left (\dfrac \right )&= ktv \qquad \text\ -\ln(2)&= kt \qquad \text\ -\ln(2)k&= t \qquad \text \end\] Since \(t\), the time, is positive, \(k\) must, as expected, be negative.

Expressed in scientific notation, this is \(4.01134972 × 1013\).

In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

In real-world applications, we need to model the behavior of a function.

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.

Again, we have the form \(y=A_0e^\) where \(A_0\) is the starting value, and \(e\) is Euler’s constant.

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