# Doubling time practice for fun and profit

Here is some practice in finding doubling times:

### A strep culture is sampled at two points in time, 1 hour and 10 minutes apart. 3.2 x 10^{6} cells/ml are found in the first sample, and 1.8 x 10^{7} cells/ml in the second sample. What was the doubling time?

- I need a hint ... : try estimating the doubling time first -- notice that there were between 2 and 3 generations in 70 minutes...

- ...another hint ... : when you plug the numbers into the formula, you get 1.8 x 10
^{7}= 3.2 x 10^{6}* 2^{g}.

- ...another hint ... : dividing both sides by 3.2 x 10
^{6}will make the equation less complicated - ...another hint ... : Starting with 5.625 = 2
^{g}, get g out of the exponent by taking the log of both sides - ...another hint ... : log(5.625) = g log(2), so what's g?
- ...another hint ... : g (the number of generations) was 2.49, which took 70 minutes, so what's the generation time?

#### I think I have the answer: 70/2.49 = 28.1 minutes.

### A beginning yogurt maker decides to test how consistently his yogurt cultures grow. He starts with 1 x 10^{5} cells/ml in each of two cultures. After 4 hours, he has 16 x 10^{5} cells/ml in the first culture, and 20 x 10^{5} cells/ml in the second culture. Are the doubling times within 10 minutes of each other?

- I need a hint ... :Doubling time for the first culture is easy to figure out even without logs.

- ...another hint ... : make sure you state both times in minutes, not hours

- ...another hint ... : Since the second culture has more cells/ml, its doubling time must be lower.

#### I think I have the answer: Yes, the times are 60 and 55.5 minutes.

### A syphilis patient has tissue cultures taken one week apart. The first sample has 5000 CFUs per ???, while the second sample has 32000 CFUs. Four days later a third sample has 58000 CFUs. Is the population growing exponentially?

- I need a hint ... : what is the doubling time in the first week?

- ...another hint ... : what is the doubling time in the next 4 days?

- ...another hint ... : remember that exponential growth means constant doubling time

#### I think I have the answer: No -- Doubling time was faster for the first week (2.61 days) compared to the next four days (4.66 days).

### A doctor receives a pathogen sample which she believes to be M. tuberculosis (doubling time 13 hours in culture). She cultures the sample, finding that it grows from 0.2x10^{5} cells/ml to 8 x 10^{6} cells/ ml in 24 hours. Assuming that the culture was growing exponentially, is the sample likely to be M. tuberculosis?

- I need a hint ... : In 24 hours, the cell culture should almost quadruple

- ...another hint ... : be careful with that scientific notation!

- ...another hint ... : what was the doubling time?

#### I think I have the answer: No -- the culture grew 400-fold in 24 hours-- a doubling time of a little under 3 hours.

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