Chapter 7.7, Problem 23E

Challenge Yourself

Living plants and animals all contain the chemical element carbon. A certain percentage of that carbon is radioactive, and scientists believe that the parentage has remained constant for thousands of years. Radioactive carbon decays, so that when an animal dies, a tiny bit of the radioactive carbon is lost each year. It is known that the amount of radioactive carbon that remains in a fossil at the end of a year is approximately 0.99988 of the amount that was present at the beginning. Thus, the following dynamical system describes radioactive carbon

decay in a fossil:

$C_{n+1}=0.99988·C_n$ for $n=0, 1, 2, 3,\dots$

This system behaves exactly like the compound interest situation (except the amount or radioactive carbon is decreasing), so it is easy to see that after k years, the amount of radioactive carbon in the fossil will be $C_k={0.99988}^k·C_0$. We will always assume that the amount of radioactive carbon at time 0 will be 1; that is, $C_0=1$. Use this model to answer Exercise 23 and 24. You can solve these equations using the log function, as you did in Section 7.4.

Carbon dating. A fossilized bone is found that contains 90% of the original radioactive carbon that was present. To the nearest 100 years, how old is the bone?