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*(1 + *r*)^{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(1.05)^{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* · 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*(*t*) = 100 · 2^{t}

* P*(12) = 100 · 2^{12} = 409,600 bacteria