The US Department requires that pasteurized milk contain no more than 20,000 bacteria per milliliter. It has been established that the number of bacteria in milk stored at 4.5 ° C can double in 39 hours. If after pasteurization, a sample of milk contains 20,000 bacteria per milliliter, what will be the number of bacteria per milliliter after 10 days?

Since we know the number of bacteria per milliliter doubles each 39 hours, we expect it to be a exponential growth, with 2 being the exponent (because it always doubles at a definite time interval). Such a problem can be solved by an equation like this one:

[tex]N_t=N_0\times2^{t/T}[/tex]

In the equation above, Nt represents the number of bacteria per milliliter at time t, whereas N0 is the initial amount of bacteria per milliliter. T represents the time it takes for the number of bacteria per milliliter to double.

Step 2 - Substitute the numerical values and work the math

According to the exercise, t = 240 h (10 days), T = 39 h, N0 = 20000 bacteria per milliliter. Substituting these values on the equation above:

[tex]N_t=20000\times2^{240/39}[/tex]

Working the numbers, we have:

[tex]N_t=20000\times2^{6.1}=20000\times68.6=1.3\times10^6\text{ bacteria per milliliter}[/tex]

So, after 10 days, there will be aproximmately 1 billion bacterias per milliliter.