Suppose that the population of a colony of bacteria is growing exponentially. at the start of an experiment, there are 4000 bacteria. two hours later there are 4600 bacteria. how long will it take the colony to reach 8200 bacteria? : *

It takes 10.3 hours to reach 8200 bacteria

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).

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)

Now we can combine the two formulas:

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  

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