College Algebra & Trigonometry, 2018a

3.5. APPLICATIONS OF THE NEGATIVE EXPONENTIAL FUNCTION 179
3.5 Applications of the Negative Exponential Function
At the beginning of Chapter 3, we worked with application problems and solved
them using the graphing calculator. In this section, we will revisit some of these
application problems and use the solution methods discussed in the previous
sections to solve these problems algebraically.
Radioactive Decay
The decay of a radioactive element into its non-radioactive form occurs following
a time line dictated by the ”half-life” of the element. The half-life is the amount
of time that it takes for half of the existing radioactive material to decay to its
non-radioactive form.
Consider the equation:
A(t) =A 0 e −kt
where A(t) is the amount of material left at time t, A 0 is the amount present at
t =0, and k is a constant that can be determined based on the half-life of the
material.
If we know that after one half-life, there will 50% of the radioactive material remaining,
then we can say that:
0.5A 0 = A 0 e −kt h
where t h is the half-life. To solve this equation for k, we would first divide on
both sides by A 0 :
0.5A 0
A 0
= A 0e −kt h
A 0
0.5 =e −kt h