Respuesta :
Answer:
[tex]P(t) = 100e^{0.0251t}[/tex]
The doubling time is of 27.65 minutes.
Step-by-step explanation:
Exponential equation of growth:
The exponential equation for population growth is given by:
[tex]P(t) = P(0)e^{kt}[/tex]
In which P(0) is the initial value and k is the growth rate.
A freshly inoculated bacterial culture of Streptococcus contains 100 cells.
This means that [tex]P(0) = 100[/tex]. So
[tex]P(t) = 100e^{kt}[/tex]
When the culture is checked 60 minutes later, it is determined that there are 450 cells present.
This means that [tex]P(60) = 450[/tex], and we use this to find k. So
[tex]450 = 100e^{60k}[/tex]
[tex]e^{60k} = 4.5[/tex]
[tex]\ln{e^{60k}} = \ln{4.5}[/tex]
[tex]60k = \ln{4.5}[/tex]
[tex]k = \frac{\ln{4.5}}{60}[/tex]
[tex]k = 0.0251[/tex]
So
[tex]P(t) = 100e^{0.0251t}[/tex]
Doubling time:
This is t for which P(t) = 2P(0) = 200. So
[tex]200 = 100e^{0.0251t}[/tex]
[tex]e^{0.0251t} = 2[/tex]
[tex]\ln{e^{0.0251t}} = \ln{2}[/tex]
[tex]0.0251t = \ln{2}[/tex]
[tex]t = \frac{\ln{2}}{0.0251}[/tex]
[tex]t = 27.65[/tex]
The doubling time is of 27.65 minutes.
