Answer :
ANSWER:
(a)
[tex]p=72\cdot e^{-0.472t}[/tex](b)
During the 9th week, in week 10, the population has already disappeared
STEP-BY-STEP EXPLANATION:
We have that the function has the following form:
[tex]p=a\cdot e^{kt}[/tex](a)
We can calculate the value of k, knowing that after two weeks the population dropped from 72 to 28, therefore we replace these values and solve for k, like this:
[tex]\begin{gathered} 28=72\cdot e^{k\cdot2} \\ e^{2k}=\frac{28}{72} \\ 2k=\ln (\frac{28}{72}) \\ k=\frac{\ln(\frac{28}{72})}{2} \\ k=-0.472 \end{gathered}[/tex]Therefore, the function would be:
[tex]p=72\cdot e^{-0.472t}[/tex](b)
In order to predict when the population is less than 1, we must do the following inequality:
[tex]\begin{gathered} 1<\: 72\cdot\: e^{-0.472t} \\ \frac{1}{72}\frac{1}{72} \\ -0.472t>\ln (\frac{1}{72}) \\ t<\frac{\ln (\frac{1}{72})}{-0.472} \\ t<9.05 \end{gathered}[/tex]Which means that between the first 8 weeks the population will be greater than 1 and during the ninth week the population will begin to be less than 1
