In a certain culture of bacteria, the number of bacteria increased sixfold in 10h. Assuming natural growth, how long did it take for their number to double?
This is classic exponential growth so use the equation A(t) = A_0e^{kt}
where:
- $\displaystyle A(t)$ = Amount at time t
- $\displaystyle A_0$ = Amount at time 0/initial amount
- $\displaystyle k$ = growth constant
- $\displaystyle t$ = time
By definition the amount at time 0 is equal to $\displaystyle A_0$
If the increase is sixfold after ten hours we get the following:
$\displaystyle 6A_0 = A_0e^{10k}$ which is equal to $\displaystyle 6 = e^{10k}$
From there you can find k
Once you know your value of k find t using the rearranged equation below and that $\displaystyle A_2 =2A_0= A_0e^{kt}$
$\displaystyle t = \frac{1}{k} \left[ \ln (A_t) - \ln (A_0)\right]$