Question

The bacterium, cholera, reproduces at an exponential rate which can be modeled through the continuous growth model. If the value for k is 1.38 and we initially start with 2 bacteria, what is the equation for cholera growth, where t is the number of hours?

Answer 1

The continuous growth rate model stipulates that the rate of population change $\dfrac{\mathrm dP}{\mathrm dt}$ is in proportion with the actual population $P(t)$ (a function of time $t$). That is, there is some $k$ such that

$\dfrac{\mathrm dP}{\mathrm dt}=kP$

Solving for $P$, you get

$\dfrac{\mathrm dP}{P}=k\,\mathrm dt\implies \ln|P|=kt+C\implies P=Ce^{kt}$

You're given that $k=1.38>0$, so the population is increasing. At time $t=0$, you start with 2 bacteria, so $P(0)=2$ and

$2=Ce^{k\times0}\implies C=2$

So, the growth equation is

$P(t)=2e^{1.38t}$