Answer:
It will take 1.7 hours for the colony to contain 2,000 bacteria.
Explanation:
One bacteria divides into two by the process of binary fission.
Initial bacteria population = 200
Growth factor = 2
It doubles in size every 30 minutes.
Time = t/30
The exponential growth function is
[tex]y=ab^x[/tex]
where, a is initial value, b is growth factor and x is time.
Substitute a=200, b=2 and [tex]x=\frac{t}{30}[/tex] in the above function.
[tex]y=200(2)^{\frac{t}{30}[/tex]
We need to find the time taken by bacteria to reach 2,000 bacteria.
Substitute y=2000 in the above equation.
[tex]2000=200(2)^{\frac{t}{30}[/tex]
Divide both sides by 200.
[tex]10=(2)^{\frac{t}{30}}[/tex]
Taking log both sides.
[tex]\log 10=\log (2)^{\frac{t}{30}}[/tex]
[tex]1=\frac{t}{30}\log (2)[/tex]
[tex]30=\log 2(t)[/tex]
Divide both sides by log 2.
[tex]\dfrac{30}{\log 2}=t[/tex]
[tex]\dfrac{30}{0.301}=t[/tex]
[tex]99.667774=t[/tex]
It will take 99.66774 minutes for the colony to contain 2,000 bacteria.
1 hour = 60 minute
[tex]t=\dfrac{99.667774}{60}=1.66112\approx 1.7[/tex]
Therefore, it will take 1.7 hours for the colony to contain 2,000 bacteria.
