The growth rate of Escherichia coli, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 30 min.

Question

(b) How long would it take for a colony of 50 cells to increase to a population of 1 million? (Round your answer to the nearest whole number.)

Answer 1

we have already done a half life one but anyway

ds/dt = k s

ds/s = k dt

ln s = k t + c

when t = 0, ln si = c
si is initial size

ln s = k t + ln si

ln (s/si) = k t

s/si = e^kt

2 = e^k(30 min)
ln 2 = 30 k
k = .0231

1,000,000/50 = e^.0231 t
ln (20,000) = .0231 t
t = 428.6 = 429 minutes
about 7 1/4 hours

Answer 2

Thinking of the doubling rate, we can see that after t hours, the population is

P(t) = 50 * 2^(2t) = 50*4^t
So, we want

50*4^t = 1000000
4^t = 20000
t log4 = log 20000
t = log20000/log4 = 7.14 hours