EXAMPLE 7 How Old?

We turn the question around to determine the age from the observed number of decays: How old is a sample that is decaying at a rate of 105 grams per hour per gram of carbon?

The formula that relates , the age of the sample in years, and , the number of carbon-14 atoms disintegrating per gram per hour, is

Solving for gives

Using natural logarithms, we can solve for as

Thus, a sample decaying at a rate of 105 atoms per hour per gram of carbon would be years old.