8. a. Let y(t) be the amount of a radiactive material with relative decay rate k. Let Q(t) be the decay rate. Use the differential equation for y (not the solution formula) to show that the quantity Q also undergoes exponential decay with rate constant k.
b. It is sometimes easier to measure the rate of radioactive decay than the amount of material. In example 4, the composition of the sample is taken as given. The same date can be obtained without knowing the amount of material, by making use of the result of part a. To see this, first calculate the decay rate Q(t) = -y'(t), using the solution of Example 4. Then determine the values of Q corresponding to time Q and to the current time t = 1.14 X 10^9 yr. Now assume that all you know are (1) the fact that Q decays exponentially to 0 with rate constant k = 1.537 X 10^-10 yr^-1 and (2) the initial value of Q, and (3) the current value of Q. Use this information to determine the age of the sample.
This solution shows how to determine the differential equations for the standard of the decay and the age of the sample. This solution is presented in an attached Word document.