# Exponential Growth

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