Question

1. Suppose a single bacterium lands on one of your teeth and starts reproducing by a factor of 2 every hour. Ifnothing is done to stop the growth of the bacteria, write a function for the number of bacteria as a functionofthe number of days.

Answer 1

Answer: We can model the problem by using the following formula:

$F(t)=Ie^{rt}\rightarrow(1)$

Where F is the final bacteria and I is the initial bacteria at the start, r is the doubling constant, and t is time:

$\begin{gathered} \begin{equation*} F(t)=Ie^{rt} \end{equation*} \\ \\ 4=2e^{r(1)} \\ \\ 2=e^r \\ \\ r=ln(2)=0.69314718055 \\ \\ r=0.693 \end{gathered}$

Therefore for the number of hours, the function is:

$F(t)=2e^{0.693t}\rightarrow(2)$

Similarly, for the number of days, the function is as follows:

$\begin{gathered} \begin{equation*} F(t)=Ie^{rt} \end{equation*} \\ \\ 33554432=2e^t \\ \\ r=ln(\frac{33554432}{2})=16.636 \\ \\ r=16.636 \\ \\ \therefore\rightarrow \\ \\ F(t)=2e^{16.636t}\rightarrow(3) \end{gathered}$

Equation (3) is the final function.