1. **State the problem:** We have populations of six towns given by exponential functions of the form $P = P_0 (r)^t$, where $P_0$ is the initial population and $r$ is the growth/decay factor per year. 2. **Recall the rules:** - If $r > 1$, the population is growing. - If $r < 1$, the population is shrinking. - The annual percentage growth or decay rate is calculated by $\text{Rate} = (r - 1) \times 100\%$. 3. **Analyze each town:** - Town 1: $r=0.8 < 1$ (shrinking) - Town 2: $r=0.94 < 1$ (shrinking) - Town 3: $r=1.197 > 1$ (growing) - Town 4: $r=1.16 > 1$ (growing) - Town 5: $r=1.09 > 1$ (growing) - Town 6: $r=0.78 < 1$ (shrinking) 4. **Answer (a):** Towns growing are 3, 4, 5. 5. **Answer (b):** Towns shrinking are 1, 2, 6. 6. **Answer (c):** The fastest growing town is the one with the largest $r$ above 1, which is Town 3 with $r=1.197$. Calculate its growth rate: $$\text{Rate} = (1.197 - 1) \times 100 = 0.197 \times 100 = 19.7\%$$ 7. **Answer (d):** The fastest shrinking town is the one with the smallest $r$ below 1, which is Town 6 with $r=0.78$. Calculate its decay rate: $$\text{Rate} = (1 - 0.78) \times 100 = 0.22 \times 100 = 22\%$$ 8. **Answer (e):** The largest initial population $P_0$ is Town 1 with 2500. **Final answers:** (a) 3,4,5 (b) 1,2,6 (c) Town 3, 19.7% (d) Town 6, 22% (e) Town 1