Given data:
[tex]\begin{gathered} P_0=4000 \\ t=12 \\ P=12500 \\ \\ P=P_0e^{kt} \end{gathered}[/tex]
Use the given data in the model and solve k (growth rate):
[tex]12500=4000e^{12k}[/tex]
Divide both sides of the equation into 4000:
[tex]\begin{gathered} \frac{12500}{4000}=\frac{4000}{4000}e^{12k} \\ \\ \frac{25}{8}=e^{12k} \end{gathered}[/tex]
Find the natural logarithm of both sides of the equation:
[tex]\begin{gathered} \ln (\frac{25}{8})=\ln (e^{12k}) \\ \\ \ln (\frac{25}{8})=12k \end{gathered}[/tex]
Divide both sides of the equation by 12:
[tex]\begin{gathered} \frac{\ln (\frac{25}{8})}{12}=\frac{12}{12}k \\ \\ \frac{\ln(\frac{25}{8})}{12}=k \\ \\ \\ k=\frac{\ln(\frac{25}{8})}{12} \end{gathered}[/tex]
Evaluate:
[tex]k\approx0.095[/tex]
Then, the growth rate is 0.095 (9.5%)
