Question

A culture contains 2000 bacteria initially and doubles every 40 minutes. Assuming that the rate of growth is proportional to the number of bacteria, find a function that models the number P(t) of bacteria after t minutes.

Answer 1

Answer: Hello there!

We know that the culture has 2000 bacteria initially (at t = 0 min), and it doubles when t = 40min, again at t = 80 min, and so on

And the rate of grouth is proportional to the population, then we can obtain an expression of the form:

P(t) = 2000*k(t) where k(t) is the factor that defines the growth of the population:

we know that:

p(0) = 2000*k(0) = 2000

p(40) = 2000*k(40) = 2000*2 = 4000

p(80) = 2000*k(80) = 2000*2*2 = 8000

etc

then we can see that k(t) = 2^(t/40)

where k(0) = 1, k(40) = 2, k(80) = 2*2, etc

then the function that models the population of the culture is:

p(t) = 2000*2^(t/40)

where t is in minutes.