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.
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 · 2t
with t in hours.
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 · 2t
P(12) = 100 · 212 = 409,600 bacteria