Question

a culture started with 6000 bacteria . after 5 hours it grew 7800 bacteria. predict how many bacteria will be present after 13 hours . use the formula P = Ae^kt round the nearest whole number

Answer 1

We know that we need to use the formula: $P=Ae^{kt}$
where 
$P$ is final population of bacteria after $t$ hours
$A$ is the initial population 
$k$ is the growth rate 
$t$ is the time in hours

The first thing we are going to do is find $k$. We know for our problem that: $P=7800$, $A=6000$, and $t=5$. Lets replace those values in our formula:
$P=Ae^{kt}$
$7800=6000e^{5k}$
Now, to find $k$ we are going to isolate $e^{5k}$, and then apply logarithms:
$\frac{7800}{6000} =e^{5k}$
$e^{5k}= \frac{13}{10}$
$ln(e^{5k})=ln( \frac{13}{10} )$
$5k=ln( \frac{13}{10})$
$k= \frac{ln( \frac{13}{10}) }{5}$
$k=0.052$

Now that we have $k$, we are going to use our formula one more time, but this time $t=13$:
$P=Ae^{kt}$
$P=6000e^{(0.052)(13)$
$P=6000e^{0.676}$
$P=11796$

We can conclude that the culture will have 11796 bacteria after 13 hours.