# A bacterial colony has an initial population of 500. After 3 hours, the population is estimated...

## Question:

A bacterial colony has an initial population of 500. After 3 hours, the population is estimated to be 1800. What would you expect the population to be after 10 hours?

## Exponential Growth:

Exponential growth refers to the way the population of a certain species changes as a function of time. Its name is derived from the mathematical model, which contains an exponential function. This model comes from the fact that the rapidity of population change is proportional to the number of individuals present. The equation that represents exponential growth is: {eq}N=N_0e^{kt} {/eq}

We know that the model of this situation is:

{eq}N=N_0e^{kt} {/eq} Equation 1

{eq}N_0=500 {/eq}

At {eq}t=3 {/eq}, {eq}N=1800 {/eq}

By replacing in Equation 1:

{eq}1800=500e^{k*3}\\ \displaystyle \frac{1800}{500}=e^{3k}\\ 3.6=e^{3k}\\ \ln(3.6)=ln(e^{3k})\\ \ln(3.6)=3k\\ \displaystyle k=\frac{\ln(3.6)}{3}\\ k=0.4270 {/eq}

The model is:

{eq}N=500e^{0.4270t} {/eq}

At {eq}t=10 {/eq}:

{eq}N=500e^{0.4270*10}\\ N=500e^{4.27}\\ N=500(71.52)\\ N=35760 {/eq}