Answer:
It takes 10.3 hours to reach 8200 bacteria
Step-by-step explanation:
Under laboratory-controlled conditions, the growth of bacteria is exponential, in mathematical terms is a geometric progression base 2, that is, one bacteria reproduce and converts in two ( one generation), then the two bacteria reproduces and now we have 4 bacteria (second generation), etc. The time that takes in doubling the growth is called generation time. We can calculate with the next formula.
Generation time (g)= time that hass passed (t) / number of generations (n).
g= t/n
We can calculate the number of generations by the next formula:
Number of generations (n)= logarithm of number final of cells (log N) – logarithm of the number of initial cells ( log No) divided in the logarithm of two ( log 2)
n= 3.3 (log N – log Nо)
Now we can combine the two formulas:
g= t / 3.3( log N – log Nо )
First, we need to know the generation time of that bacteria.
Inicial number (N): 4000
Final number (Nо): 4600
Time passed (t): 2 hours
Generation time (g) : ?
g= 2 hours / 3.3 ( log 4600 – log 4000)
g= 2 hours / 3.3 ( 3.66 – 3.6 )
g= 2 hours / 3.3 (0.06)
g= 2 hous/0.198 generations
g= 10.1 hours of generation time
The bacteria takes 10.1 hours in doubling the time. How much time does it need to reach 8200 bacteria?
We know:
Inicial number (N): 4000
Final number (Nо): 8200
Times passed (t) : ?
Generation time (g): 10.1 hours
With the same formula,
g= t / 3.3 ( log N – log Nо)
10.1 = t / 3.3 ( log 8200 – log 4000)
10.1 = t / 3.3 ( 3.91 – 3.6 )
10.1 = t / 3.3 (0.31)
10.1 = t / 1.02 generations
As we want to know the time, the 1.02 generation pass on the other side multiplaying.
t= (1.02) 10.1
t= 10.3 hours