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One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications.
Many systems exhibit exponential growth. These systems follow a model of the form $y={y}_{0}{e}^{kt},$ where ${y}_{0}$ represents the initial state of the system and $k$ is a positive constant, called the growth constant . Notice that in an exponential growth model, we have
That is, the rate of growth is proportional to the current function value. This is a key feature of exponential growth. [link] involves derivatives and is called a differential equation. We learn more about differential equations in Introduction to Differential Equations .
Systems that exhibit exponential growth increase according to the mathematical model
where ${y}_{0}$ represents the initial state of the system and $k>0$ is a constant, called the growth constant .
Population growth is a common example of exponential growth. Consider a population of bacteria, for instance. It seems plausible that the rate of population growth would be proportional to the size of the population. After all, the more bacteria there are to reproduce, the faster the population grows. [link] and [link] represent the growth of a population of bacteria with an initial population of $200$ bacteria and a growth constant of $0.02.$ Notice that after only $2$ hours $(120$ minutes), the population is $10$ times its original size!
Time (min) | Population Size (no. of bacteria) |
---|---|
$10$ | $244$ |
$20$ | $298$ |
$30$ | $364$ |
$40$ | $445$ |
$50$ | $544$ |
$60$ | $664$ |
$70$ | $811$ |
$80$ | $991$ |
$90$ | $1210$ |
$100$ | $1478$ |
$110$ | $1805$ |
$120$ | $2205$ |
Note that we are using a continuous function to model what is inherently discrete behavior. At any given time, the real-world population contains a whole number of bacteria, although the model takes on noninteger values. When using exponential growth models, we must always be careful to interpret the function values in the context of the phenomenon we are modeling.
Consider the population of bacteria described earlier. This population grows according to the function $f(t)=200{e}^{0.02t},$ where t is measured in minutes. How many bacteria are present in the population after $5$ hours $(300$ minutes)? When does the population reach $\mathrm{100,000}$ bacteria?
We have $f(t)=200{e}^{0.02t}.$ Then
There are $\mathrm{80,686}$ bacteria in the population after $5$ hours.
To find when the population reaches $\mathrm{100,000}$ bacteria, we solve the equation
The population reaches $\mathrm{100,000}$ bacteria after $310.73$ minutes.
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