5. The population of a town is 15,000 . It decreases at a rate of \( 11 \% \) per year. In about how many years will the population drop below 9,000 ? \( \begin{array}{ll}\text { a) } 5 & \text { b) } 6 \\ \text { c) } 3 & \text { d) } 4\end{array} \)

To determine when the population drops below 9,000, we can use the exponential decay formula: \( P(t) = P_0 \cdot (1 - r)^t \) where: - \( P_0 \) is the initial population (15,000), - \( r \) is the rate of decrease (11% or 0.11), - \( t \) is the number of years, - \( P(t) \) is the population after \( t \) years. We set the equation to find when \( P(t) < 9,000 \): \[ 15,000 \cdot (1 - 0.11)^t < 9,000 \] Solving this gives: \[ (0.89)^t < \frac{9,000}{15,000} \] \[ (0.89)^t < 0.6 \] Taking the logarithm of both sides: \[ t \cdot \log(0.89) < \log(0.6) \] This results in: \[ t > \frac{\log(0.6)}{\log(0.89)} \approx 3.67 \] So, it will take approximately 4 years for the population to drop below 9,000. Therefore, the answer is **d) 4**. Now to dive into some additional fun facts: The decline of a population can often result in a variety of social and economic changes, like a reduction in local business or the revitalization of the area as it might become attractive for new residents. In many towns facing population decline, community involvement often spikes as citizens work towards revitalization and new development. Curiously, the concept of exponential decay isn't limited to populations! You can find it everywhere—from radioactive decay to how quickly your leftovers might go bad in the fridge. So, the next time you watch a movie about time travel, you might just think about how fast or slow things decay in terms of population and beyond!