Given:
The population of the bacteria at the beginning = 80
the population grows according to a continuous exponential growth model.
after 14 days, there are 216 bacteria.
y = the number of bacteria after time t
So, the general relation between y and t will be:
[tex]y=a\cdot e^{bt}[/tex]
We need to find the values of a and b
At t = 0 y = 80
So,
[tex]\begin{gathered} 80=a\cdot e^0 \\ a=80 \end{gathered}[/tex]
When t = 14 , y = 216
So,
[tex]\begin{gathered} 216=80e^{14b} \\ \frac{216}{80}=e^{14b} \\ \text{2}.7=e^{14b} \\ \ln 2.7=14b \\ \text{0}.99325=14b \\ b=\frac{0.99325}{14}=0.071 \end{gathered}[/tex]
so, the function will be:
[tex]y=80\cdot e^{0.071t}[/tex]
Part b: we need to find the number of bacteria after 23 days
So, substitute with t = 23
so,
[tex]y=80\cdot e^{0.071\cdot23}=409[/tex]
so, after 23 days the number of bacteria = 409
