Chapter 12.3, Problem 46E

Radioactive decay Radioactive substances decay at a rate that is proportional to the amount present. Thus, if k is a constant and the amount present is x, the decay rate is

$\frac{dx}{dt}=kx\left(t\text{ in hours}\right)$

This means that the relationship between the time and the amount of substance present can be found by evaluating the integral

$t={\displaystyle \int \frac{dx}{kx}}$

(a) Evaluate the integral to find the relationship.

(b) Use properties of logarithms and exponential functions to write x as a function of t.