Suppose a bacteria culture grows at a rate proportional to t

Suppose a bacteria culture grows at a rate proportional to the number of cells present. A colony of bacteria is observed to increase in size by 30% over a 4-week period. How long will the colony take to triple its initial size?

Solution

It will be an exponential curve

dPo/dt = kPo

ln(Pt/Po) = kt

Pt = Po * e^(kt)

after 4 week, Pt = P0 + 30/100 * Po = 1.3Po

1.3Po = Po * e^(k * 4)

4k = ln(1.3)

k = ln(1.3)/4

Now time taken for the population to be tripled will be

3Po = Po * e^(kt)

kt = ln(3)

t = 4ln(3)/ln(1.3) = 16.749 weeks