For this problem, we know the size of a certain population and the rate of decay on its numbers. We need to determine the time it will take for the population to reach 1500 people.
The population can be modeled with the following equation:
[tex]P(t)=12000\cdot(0.85)^t[/tex]Where "t" is the number of years and P is the population. We want to find how long it will take before the population is equal to 1500, so we need to replace P with 1500 and solve for t. We have:
[tex]\begin{gathered} 1500=12000\cdot(0.85)^t\\ \\ 0.85^t=\frac{1500}{12000}\\ \\ 0.85^t=0.125\\ \\ \ln0.85^t=\ln0.125\\ \\ t=\frac{\ln0.125}{\ln0.85}=\frac{-2.07944}{-0.162519}=12.79505 \end{gathered}[/tex]It will take approximately 12.8 years to reach 1500 people.
