Posted by **Erica** on Thursday, October 19, 2006 at 12:18pm.

A culture of bacteria obeys the law of unlimited growth. If 500 bacteria are present initially and there are 800 after 1 hour, how many will be present in the cultrue after 5 hours? How long until there are 20,000 bacteria?

Hmmm, what is "the law of unlimited growth"? In any case, this looks like exponential growth to me.

The most basic formula for exponential growth is

A = P*e^{r*t} where r is the rate of growth, t is the time, P is the initial amount and A is observed growth.

Here 800 = 500*e^{r*1hr} so

ln(1.6units) = r*1hr or ln(1.6units)/1hr = r or

A = 500*(1.6)^{t}

Substitute 5 for t to answer the first question, i.e.

A = 500*(1.6)^{5}

To answer the second part set A=20,000

20,000=500*(1.6)^{t} or

40 = (1.6)^{t}

It looks to be slightly less than 8 hrs. You can verify this.