# A population of bacteria is growing in such a way a that the number of bacteria present, N, after t minutes is given by the rule N=42e^(0.0134t). How long will it be before the population bacteria doubles?

Mar 30, 2017

51.73 minutes.

#### Explanation:

1. Let the population be $N$ at $t = {t}_{1}$.
2. Let the population be twice i.e. $2 \cdot N$ at $t = {t}_{2}$.

$\therefore \delta t = {t}_{2} - {t}_{1}$ where $\delta t$ is the time required for population to double.

$\therefore$ we obtain two equations from above conditions:-

1. $N = 42 \cdot {e}^{0.0134 \cdot {t}_{1}}$
2. $2 \cdot N = 42 \cdot {e}^{0.0134 \cdot {t}_{2}}$

Dividing these equations

$2 = {e}^{0.0134 \cdot \left({t}_{2} - {t}_{1}\right)} = {e}^{0.0134 \cdot \delta t}$

Taking natural logarithm on both sides;
$\ln \left(2\right) = 0.0134 \cdot \delta t$
$\implies$$\delta t = 51.73$