The discussion revolves around determining the initial population of bacteria in a culture, given that the population increases at a rate proportional to its current size. Participants explore the mathematical modeling of this growth using exponential functions and derive equations based on population changes over specific time intervals.
Participants generally agree on the approach to model the population growth and the equations involved, but there is no consensus on the specific values of $$P_0$$ and $$k$$ as they remain to be determined through further calculations.
The discussion includes algebraic manipulations that depend on the assumptions made about the form of the population growth model and the values of $$k$$ and $$P_0$$, which have not yet been resolved.
MarkFL said:We know the population will take the form:
$$P(t)=P_0e^{kt}$$
We are then given:
$$P(3)-P(2)=2455$$
$$P(5)-P(4)=4314$$
This will give you 2 equations and 2 unknowns...can you proceed to find the initial population $P_0$?
rayne said:What would be the k value?
MarkFL said:This would also have to be algebraically determined. Let's look at the first equation I gave:
$$P(3)-P(2)=2455$$
This then becomes (using the definition of $P(t)$):
$$P_0e^{3k}-P_0e^{2k}=2455$$
Now, if we factor on the left, we obtain:
$$P_0e^{2k}\left(e^k-1\right)=2455$$
And solving for $P_0$ we then get:
$$P_0=\frac{2455}{e^{2k}\left(e^k-1\right)}$$
Now, if we state the second equation I gave, we have (after factoring):
$$P_0e^{4k}\left(e^k-1\right)=4314$$
At this point we may substitute for $P_0$ that we obtained above:
$$\frac{2455}{e^{2k}\left(e^k-1\right)}e^{4k}\left(e^k-1\right)=4314$$
Now you may simplify, then solve for $e^k$, and then you can determine $P_0$.