Respuesta :
Answer:
a) k=2.08 1/hour
b) The exponential growth model can be written as:
[tex]P(t)=Ce^{kt}[/tex]
c) 977,435,644 cells
d) 2.033 billions cells per hour.
e) 2.81 hours.
Step-by-step explanation:
We have a model of exponential growth.
We know that the population duplicates every 20 minutes (t=0.33).
The initial population is P(t=0)=58.
The exponential growth model can be written as:
[tex]P(t)=Ce^{kt}[/tex]
For t=0, we have:
[tex]P(0)=Ce^0=C=58[/tex]
If we use the duplication time, we have:
[tex]P(t+0.33)=2P(t)\\\\58e^{k(t+0.33)}=2\cdot58e^{kt}\\\\e^{0.33k}=2\\\\0.33k=ln(2)\\\\k=ln(2)/0.33=2.08[/tex]
Then, we have the model as:
[tex]P(t)=58e^{2.08t}[/tex]
The relative growth rate (RGR) is defined, if P is the population and t the time, as:
[tex]RGR=\dfrac{1}{P}\dfrac{dP}{dt}=k[/tex]
In this case, the RGR is k=2.08 1/h.
After 8 hours, we will have:
[tex]P(8)=58e^{2.08\cdot8}=58e^{16.64}=58\cdot 16,852,338= 977,435,644[/tex]
The rate of growth can be calculated as dP/dt and is:
[tex]dP/dt=58[2.08\cdot e^{2.08t}]=120.64e^2.08t=2.08P(t)[/tex]
For t=8, the rate of growth is:
[tex]dP/dt(8)=2.08P(8)=2.08\cdot 977,435,644 = 2,033,066,140[/tex]
(2.033 billions cells per hour).
We can calculate when the population will reach 20,000 cells as:
[tex]P(t)=20,000\\\\58e^{2.08t}=20,000\\\\e^{2.08t}=20,000/58\approx344.827\\\\2.08t=ln(344.827)\approx5.843\\\\t=5.843/2.08\approx2.81[/tex]
