## Saturday, April 24, 2010

### Calculating Bacteria Population?

Let's assume that bacteria can reproduce once every 20 minutes during binary fission. Assume further that his rate of reproduction can be maintained. Now, calculate how many bacteria there would be after 1-hour period if you started with a single bacterium

Calculating Bacteria Population?
Bacterial growth

In this section we will return to the questions posed in the first section on exponential and logarithmic functions. Recall that we are studying a population of bacteria undergoing binary fission. In particular, the population doubles every three hours.

We would like to know the following:

How many bacteria are present after 51 hours if a culture is inoculated with 1 bacterium?

With how many bacteria should a culture be inoculated if there are to be 81,920 bacteria present on hour 42?

How long would it take for an initial population of 6 to reach a size of 12,288 bacteria?

http://www.biology.arizona.edu/biomath/t...

Exponential Population Growth

How many bacteria are present after 51 hours if a culture is inoculated with 1 bacterium? Use the model, N(t) = Noe kt, and assume the population doubles every 3 hours. (N(t) is the population size at time t and k is a constant.)

Tutorial

Now that we have convinced ourselves that an exponential model is appropriate, but not perfect, we will use the general model

N(t) = N0e k t (1)

where N (t) represents the population size at time t (Note: you may also use the model, N (t) = N0a t ). The question asks for the population size when t = 51. In order to use this model, we will need to figure out the values of the constants, N0 and k. At time t = 0, there is a single bacterium, therefore N0 = 1. Substituting N0 = 1 into (1) gives,

N(t) = 1 ⋅ e kt = e kt, (2)

We now use the fact that when t = 3, N(3) = 2 (the population has doubled). Substituting the point (t, N(t)) = (3, 2) into (2),

N(3) = 2 = e 3k (3)

We can now solve for k in (3) by "undoing" the exponential using the natural logarithm,

Using this value of k our model in (2) becomes,

(4)

Now that we have our model, we need to find the population size after 51 hours. Substituting t = 51 into (4) yields,

Thus, we find that after 51 hours there are 131,072 bacteria.

so if you start with 1 bacterium and replicates every 20 mins that means that in 1 hours there will be 3 replications therefore

2(3)=6

there will be 6 bacteria