Answer :
Answer:
a) The doubling time is 7.27 hours.
b) The population is 6 hours will be 3,543.
Step-by-step explanation:
The population of the bacteria is modeled by the following equation:
[tex]P(t) = P(0)e^{rt}[/tex]
In which P(0) is the initial population, P(t) is the population after t hours and r is the growth rate.
An experiment starts and the population of a bacteria culture increases by 10% in the first hour.
This means that
[tex]P(1) = 1.1P(0)[/tex]
Which helps us find r.
So
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]1.1P(0) = P(0)e^{r}[/tex]
[tex]e^{r} = 1.1[/tex]
Applying ln to both sides
[tex]\ln{e^{r}} = \ln{1.1}[/tex]
[tex]r = 0.0953[/tex]
So
[tex]P(t) = P(0)e^{0.0953t}[/tex]
a) What is the doubling time?
This is t when [tex]P(t) = 2P(0)[/tex]
So
[tex]P(t) = P(0)e^{0.0953t}[/tex]
[tex]2P(0) = P(0)e^{0.0953t}[/tex]
[tex]e^{0.0953t} = 2[/tex]
Applying ln to both sides
[tex]\ln{e^{0.0953t}} = \ln{2}[/tex]
[tex]0.0953t = \ln{2}[/tex]
[tex]t = \frac{\ln{2}}{0.0953}[/tex]
[tex]t = 7.27[/tex]
The doubling time is 7.27 hours.
b) If the initial population is 2,000, what is the population in 6 hours?
This is P(6) when [tex]P(0) = 2000[/tex]. So
[tex]P(t) = P(0)e^{0.0953t}[/tex]
[tex]P(6) = 2000e^{0.0953*6} = 3543[/tex]
The population is 6 hours will be 3,543.