Question

How many bacteria will there be?Round to the nearest whole number

Answer 1

You know that:

- She puts a population of 300 bacteria into a favorable growth medium at 8:00 A.M.

- At 5:00 P.M. the population is 1100 bacteria,

- The next morning, at 8:00 A.M. she comes back to the lab.

By definition, an Exponential Growth Model has the following form:

$P=P_0e^{rt}$

Where "r" is the growth rate (in decimal form), "t" is the number of times intervals and this is the initial amount:

$P_0$

1. In this case, from 8:00 A.M to 5:00 P.M. you know that:

$\begin{gathered} P=1100 \\ t=9 \end{gathered}$

And the initial amount is:

$P_0=300$

2. Then, you can substitute values into the equation and solve for "r", in order to find its value:

$\begin{gathered} P=P_0e^{rt} \\ \\ 1100=300\cdot_{}e^{r(9)} \end{gathered}$

$\frac{1100}{300}=e^{9r}$

Remember the following properties:

$\begin{gathered} ln(a)^m=m\cdot\log (a)^{} \\ \\ \ln (e)=1 \end{gathered}$

Then taking Natural Logarithm on both sides and simplifying, you get:

$\begin{gathered} \ln (\frac{1100}{300})=\ln (e)^{9r} \\ \\ \ln (\frac{1100}{300})=9r\cdot\ln (e) \\ \\ \ln (\frac{1100}{300})=9r(1) \\ \\ \ln (\frac{1100}{300})=9r \\ \\ \frac{\ln (\frac{1100}{300})}{9}=r \end{gathered}$

$r\approx0.1444$

3. From 8:00 A.M. to 8:00 A.M in the next morning, there are 24 hours. Then, you can say that:

$\begin{gathered} P_0=300 \\ r\approx0.1444 \\ t=24 \end{gathered}$

Now you can substitute values and find the number of bacteria at 8:00 A.M of the next morning:

$\begin{gathered} P=P_0e^{rt} \\ \\ P=300\cdot e^{(0.1444)(24)} \\ \\ P=300\cdot e^{(3.4656)} \end{gathered}$

$P\approx9599$

Hence, the answer is:

$P\approx9599$