The initial population was 234.
Explanation
Formula for the exponential growth is: [tex]A= P*e^r^t[/tex] , 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......
[tex]337=P*e^5^r .................................. (1)\\ \\ 699=P*e^1^5^r ................................. (2)[/tex]
Now dividing equation (2) by equation (1) , we will get .......
[tex]\frac{699}{337}=\frac{e^1^5^r}{e^5^r} \\ \\ e^1^0^r = \frac{699}{337}[/tex]
Taking 'natural log' on both sides.........
[tex]ln (e^1^0^r) = ln (\frac{699}{337})\\ \\ 10r= 0.7295....\\ \\ r= 0.07295.... \approx 0.073[/tex]
Now, plugging this [tex]r=0.073[/tex] into equation (1), we will get......
[tex]337= P*e^5^(^0^.^0^7^3^)\\ \\ 337= P*1.4405 \\ \\ P= 233.946 \approx 234[/tex]
So, the initial population was 234.
