Chapter 3.6, Problem 60E

Solution

To discuss: The population growth of humans or other animals, bacterial growth, radioactive decay and compounded interest.

The problem situation requires an exponential function $y=a{e}^{-bx}$ for positive $a\text{ and }b$ .

Discussion:

All of these applications include some kind of exponential function, which is their main thing in common. This implies that the quantity we have at a given time $t$ is dependent upon the quantity at a previous time $t-1$ .

For instance, if every single bacterium splits into two new ones, we will then have two more bacteria than we did before they split. Our situation with radioactive decay is comparable. We will have half as much after a half-life, but the precise mass depends on how much there was beforehand.

The types of exponential functions that are used could be the source of the variations. Depending on factors like the availability of food, the presence of predators, the presence of natural hazards, and other factors, we may apply an exponential growth or logistic growth function for populations of animals, bacteria, and other organisms. We can consider other considerations for people, such as family planning.

On the other hand, radioactive decay would require an exponential decay function, i.e., $y=a{e}^{-bx}$ for positive $a\text{ and }b$ .