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
`(\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


`(\mathrm dP)/(\mathrm dt)=kP`

Solving for
`P`, you get


`(\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*0)\implies C=2`

So, the growth equation is


`P(t)=2e^(1.38t)`