Answer:
[tex]P(t)=256*e^{0.068185t}[/tex]
10 minutes for the population to double.
Step-by-step explanation:
The general formula for the population equation, P(t), is:
[tex]P(t)=A*e^{nt}[/tex]
We are given that for t =5 minutes and t = 2 minutes:
[tex]P(5)=360=A*e^{5n}\\P(20)=1,000=A*e^{20n}\\\\[/tex]
Solving for A and n:
[tex]ln(360)=ln(A)+5n\\ln(1,000)=ln(A)+20n\\ln(1,000)-4ln(360)=ln(A)-4ln(A)+20n-20n\\ln(A)=5.5455536\\A=256\\n=\frac{ln(360)-ln(256)}{5}\\n=0.068185[/tex]
The exponential equation that represents this situation is:
[tex]P(t)=256*e^{0.068185t}[/tex]
The population will double when P(t) = 512 bacteria:
[tex]512=256*e^{0.068185t}\\ln(2)=0.068185t\\t=10.17\ minutes[/tex]
To the nearest minute, it takes roughly 10 minutes for the population to double.
