Question

A biologist recorded a count of 337 bacteria present in a culture after 5 minutes and 699 bacteria present after 15 minutes.

Answer 1

The initial population was 234.

Explanation

Formula for the exponential growth is:   $A= P*e^r^t$ , where P is the initial amount, A is the final amount, r is the rate of growth and  t  is the time duration.

There was 337 bacteria after 5 minutes and 699 bacteria after 15 minutes. So, the equations will be......

$337=P*e^5^r .................................. (1)\\ \\ 699=P*e^1^5^r ................................. (2)$

Now dividing equation (2) by equation (1) , we will get .......

$\frac{699}{337}=\frac{e^1^5^r}{e^5^r} \\ \\ e^1^0^r = \frac{699}{337}$

Taking 'natural log' on both sides.........

$ln (e^1^0^r) = ln (\frac{699}{337})\\ \\ 10r= 0.7295....\\ \\ r= 0.07295.... \approx 0.073$

Now, plugging this $r=0.073$ into equation (1), we will get......

$337= P*e^5^(^0^.^0^7^3^)\\ \\ 337= P*1.4405 \\ \\ P= 233.946 \approx 234$

So, the initial population was 234.