The doubling period of a bacterial population is 10 minutes. At time t = 80 minutes, the bacterial population was 80000.

Question

Find the size of the bacterial population after 5 hours.

Answer 1

Let p(t) = bacterial population at time t (in minutes) Then p(t)=p(0)*2^(t/10) where p(0) is the initial population at time = 0. As a test, consider that at t=0, p(0)=p(0)*2⁰ Because 2⁰ is 1, this appears correct. As a second check, we note that p(10)=p(0)*2 which is what we expect--the population has doubled in 10 minutes--and, similarly, p(20)=p(0)*2² which is also what we expect--the population is 4 times its original size after 20 minutes.

We are told that p(80)=80,000. Therefore we can solve for p(0) as follows:

p(80)=80,000=p(0)*2^(80/10) p(0)=80,000/2⁸ Therefore p(0)=312.5. [We'll leave to the biologists how we can have 0.5 bacterium.]. After 5 hours (300 minutes),the population = p(300)=p(0)*2³⁰= 312.5*2³⁰=335.5 billion (approximately).