Stephen L. answered • 08/17/19

Recent Engineering Graduate from Bucknell University

Great Question!

If we are assuming that decreases according to the exponential model, we can use the equation dP/dt =kP where P represents the population size and dP/dt represents the rate of population growth. We can solve this equation for P by separating the variables and integrating both sides of the equation. Doing so gives us P(t)=Ce^{kt} or P(t)=P_{0}e^{kt }(setting P=P_{0} when t=0 lets us know that P_{0 }= C).

Now that we have this equation, we can use the conditions in this problem to figure out when the population will reach 2530. We know that the bacteria has an initial population of 10,000 which is another way of saying P_{0}=10000 (plug 0=t and 10000=P(0) into P(t)=P_{0}e^{kt }and you should get this).

We also know that the population is at 4000 once 4 hours have passed. We can use this to solve for k:

P(4)=4000=10000e^{k(4) }--> 2/5 = e^{k(4) }--> ln(2/5) = 4k --> k=ln(2/5)/4 --> k=-0.22.

Now, that we have solved for all the variables, we can answer the question.

P(t) = 10000e^{-0.22t}---> P=2530=10000e^{-0.22t }---> 0.253 = e^{-0.22t }---> ln(0.253) = -0.22t ---> ln(0.253)/-0.22 =t

t=6.25

So, the population will reach 2530 in 6.25 hours.