An exponentially growing bacterial population is said to increase its number from 100 to 10^7 cells in 4 hours. If this is the case, how many generations have already gone by?

Question

Aside from the number of generations, can you help me with the formula on how to get the following:

1. What is the growth rate of the population?
2. How long will it take for the population to reach 108 cells?
3. What will the final population be after 10 hrs?

Answer 1

The set up involves the equation P_F=P₀*e^R*t

Where P_f is the final population, P₀ is the initial population or population at t=0. e is Euler's number or 2.718... which is used for exponential growth or decay, R is the growth rate, and t is the time in hours.

What is the growth rate of the population?

Plugging in the numbers provided you get 10⁷=10²*e^4R ,

First dividing: 10⁷/10² = 10⁵ = e^4R,

Then using the natural log to remove the exponential function: ln(10⁵) = ln (e^4R)

Simplifying this brings down the parameters in the exponential giving 11.51 = 4R

Dividing by 4 gives you the growth rate:

R=2.88 1/hours the units for this rate are t^-1 so hours^-1

How long will it take for the population to reach 108 cells?

Now that you know the growth rate is of this species of bacteria you can answer the other questions using the general equation:

P_F=P₀*e^2.88*t

For this question the equation is 108=100*e^2.88t

This gives ln(108/100)/2.88=t

Time to reach 108 cells = 0.027 hours or 96 seconds.

What will the final population be after 10 hrs?

Plugging into the equation in the same way gives:

P_F=100*e^2.88*10

Simplifying gives the answer:

P_F=3.22*10¹⁴ cells