Question

The population of a bee hive is growing with a monthly percentage rate compounded continuously. The population doubles in $3$ months.

Answer 1

Percentage rate of the exponential growth:

The monthly percentage rate according to the exponential growth function is $\frac{\ln 3}{3}$

Use exponential function, as given below :

${\mathrm{P}}_{\left(\mathrm{t}\right)}={\mathrm{P}}_{\left(0\right)}{\mathrm{e}}^{\mathrm{rt}}$

1. ${\mathrm{P}}_{\left(0\right)}$ is the initial population
2. If the initial population doubles in $3$months, the final population will be $3{\mathrm{P}}_0$

Substitute the given parameters into the formula to get the rate "$\mathrm{r}$" :

$\begin{array}{rl}& P_{\left(t\right)}=P_{\left(0\right)}{e}^{rt}\\ & \Rightarrow 3P_{\left(0\right)}=P_{\left(0\right)}{e}^{rt}\\ & \Rightarrow 3=e^{3r}\\ & \Rightarrow \ln 3=3r\\ & \Rightarrow r=\frac{\ln 3}{3}\\ & \Rightarrow r=0.366\\ & \Rightarrow r=36.6\mathrm{\%}\end{array}$

Hence, the monthly percentage rate to the exponential growth function is $36.6\%$.