Exercise 8

The mass of a radioactive sample decays at a rate that is proportional to its mass.

1. Express this fact as a differential equation for the mass $M(t)$ using $k$ for the constant of proportionality.

2. If the initial mass is $M_0$, find an expression for the mass $M(t)$.

3. The half-life of the sample is the amount of time required for half of the mass to decay. Knowing that the half-life of Carbon-14 is 5730 years, find the value of $k$ for a sample of Carbon-14.

4. How long does it take for a sample of Carbon-14 to be reduced to one-quarter its original mass?

5. Carbon-14 naturally occurs in our environment; any living organism takes in Carbon-14 when it eats and breathes. Upon dying, however, the organism no longer takes in Carbon-14. Suppose that you find remnants of a pre-historic firepit. By analyzing the charred wood in the pit, you determine that the amount of Carbon-14 is only 30% of the amount in living trees. Estimate the age of the firepit.