Exercise 6 Page 303

The natural base $e$ has many important applications in real-life situations. We can use it to model continuously compounded interest and decay rates. This last one allows us to date any object containing organic material. Let's talk a bit about these applications.

Continuously Compounded Interest

We can model a situation for which an interest rate is compounded so frequently that the time period in between is mathematically zero. To do this, we use the formula shown below. In this formula $A$ is the amount in the account after $t$ years, $P$ is the principal, and $r$ is the annual interest rate expressed as a decimal.

Decay Rate and Carbon-14 Dating

We can use one of carbon's isotopes (a variant of the element with different numbers of neutrons) to date any sample that comes from a living organism. To model the decay rate of a radioactive isotope and find the number of atoms left $N$ after a time of decay $t,$ we use the formula shown below. In this equation the $N_0$ is the original number of atoms, and $\lambda$ is a constant depending on the isotope. Since we can estimate the abundance of the carbon isotopes in the atmosphere, by measuring its quantity in the sample we can find the time that has passed since the source was alive and breathing it from the atmosphere.