1. **Stating the problem:** We are given population data over time and need to find an equation modeling this growth. 2. **Observing the data:** The population increases from 35.0 to 75.6 thousand over 10 years, suggesting exponential growth. 3. **Exponential growth model:** The general form is $$P(t) = P_0 \cdot d^t$$ where: - $P(t)$ is the population at time $t$, - $P_0$ is the initial population, - $d$ is the growth factor per year, - $t$ is the time in years since 1996. 4. **Identify initial population:** At $t=0$, $P(0) = 35.0$ thousand, so $P_0 = 35.0$. 5. **Find growth factor $d$:** Use a known point, for example at $t=1$, $P(1) = 37.8$. 6. **Set up equation:** $$37.8 = 35.0 \cdot d^1$$ 7. **Solve for $d$:** $$d = \frac{37.8}{35.0} = 1.08$$ 8. **Check with another point:** At $t=10$, $P(10) = 75.6$. Calculate predicted population: $$P(10) = 35.0 \cdot (1.08)^{10}$$ Calculate $(1.08)^{10}$: $$1.08^{10} \approx 2.1589$$ So, $$P(10) \approx 35.0 \times 2.1589 = 75.56$$ which matches the data closely. 9. **Final model:** $$\boxed{P(t) = 35.0 \cdot (1.08)^t}$$ --- **Why is $d$ not zero in exponential functions?** - The growth factor $d$ represents how much the population multiplies each year. - If $d$ were zero, the population would be zero for all $t > 0$, which contradicts the data. - Since the population grows, $d$ must be greater than zero and typically greater than 1 for growth. Hence, $d$ is approximately 1.08, not zero.