Exponential growth models apply to any situation where the growth is proportional to the current size of the quantity of interest.

Exponential growth models are often used for real-world situations like interest earned on an investment, human or animal population, bacterial culture growth, etc.

The general exponential growth model is

$y=C{\left(1+r\right)}^{t}$,

where $C$ is the initial amount or number, $r$ is the growth rate (for example, a $2\%$ growth rate means $r=0.02$), and $t$ is the time elapsed.

**Example 1: **

A population of $32,000$ with a $5\%$ annual growth rate would be modeled by the equation:

$y=32000{\left(1.05\right)}^{t}$

with $t$ in years.

Sometimes, you may be given a doubling or tripling rate rather than a growth rate in percent. For example, if you are told that the number of cells in a bacterial culture doubles every hour, then the equation to model the situation would be:

$y=C\cdot {2}^{t}$

with $t$ in hours.

**Example 2: **

Suppose a culture of $100$ bacteria is put into a petri dish and the culture doubles in size every hour. Predict the number of bacteria that will be in the dish after $12$ hours.

$P\left(t\right)=100\cdot {2}^{t}$

$P\left(12\right)=100\cdot {2}^{12}=409,600$ bacteria