Answer:
15595 bacteria will be present after 13 hours.
Step-by-step explanation:
Continuous population growth:
The continuous population growth model, for the population after t hours, is given by:
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(0) is the initial population and r is the growth rate.
Started with 5000 bacteria
This means that [tex]P(0) = 5000[/tex]
So
[tex]P(t) = 5000e^{rt}[/tex]
After three hours it grew 6500 bacteria:
This means that [tex]P(3) = 6500[/tex]. We use this to find r.
[tex]P(t) = 5000e^{rt}[/tex]
[tex]6500 = 5000e^{3r}[/tex]
[tex]e^{3r} = \frac{65}{50}[/tex]
[tex]\ln{e^{3r}} = \ln{\frac{65}{50}}[/tex]
[tex]3r = \ln{\frac{65}{50}}[/tex]
[tex]r = \frac{\ln{\frac{65}{50}}}{3}[/tex]
[tex]r = 0.0875[/tex]
So
[tex]P(t) = 5000e^{0.0875t}[/tex]
How many bacteria will be present after 13 hours?
This is P(13). So
[tex]P(13) = 5000e^{0.0875*13} = 15594.8[/tex]
Rounding to the nearest whole number
15595 bacteria will be present after 13 hours.
