1. **State the problem:** We have two towns, A and B, with populations growing exponentially. Town A starts with 4000 people and grows at 8% per year. Town B starts with 5000 people and grows at 6% per year. 2. **Formulas for exponential growth:** The population after $t$ years is given by $$P = P_0 (1 + r)^t$$ where $P_0$ is the initial population and $r$ is the growth rate. 3. **Calculate populations after 5 years:** - Town A: $$P_A = 4000 \times (1 + 0.08)^5 = 4000 \times 1.08^5$$ - Town B: $$P_B = 5000 \times (1 + 0.06)^5 = 5000 \times 1.06^5$$ 4. **Evaluate powers:** - $$1.08^5 = 1.4693$$ (approx) - $$1.06^5 = 1.3382$$ (approx) 5. **Calculate populations:** - $$P_A = 4000 \times 1.4693 = 5877.2$$ - $$P_B = 5000 \times 1.3382 = 6691$$ 6. **Compare populations:** - Town B has more people after 5 years. - Difference: $$6691 - 5877.2 = 813.8$$ which is less than 1000. 7. **Answer for Part A:** The size of the population in Town B will be greater than that of Town A with a difference less than 1000 people. This corresponds to option C. 8. **Part B: Find when populations are equal:** Set $$4000 (1.08)^t = 5000 (1.06)^t$$ 9. **Divide both sides:** $$\frac{4000 (1.08)^t}{5000 (1.06)^t} = 1$$ $$\frac{4000}{5000} \times \frac{(1.08)^t}{(1.06)^t} = 1$$ $$\frac{4}{5} \times \left(\frac{1.08}{1.06}\right)^t = 1$$ 10. **Isolate exponential term:** $$\left(\frac{1.08}{1.06}\right)^t = \frac{5}{4} = 1.25$$ 11. **Take natural logarithm:** $$t \ln\left(\frac{1.08}{1.06}\right) = \ln(1.25)$$ 12. **Calculate logarithms:** $$\ln\left(\frac{1.08}{1.06}\right) = \ln(1.0188679) = 0.0187$$ $$\ln(1.25) = 0.2231$$ 13. **Solve for $t$:** $$t = \frac{0.2231}{0.0187} = 11.93$$ years (approx) 14. **Calculate population at $t=11.93$:** Using Town A's formula: $$P = 4000 \times 1.08^{11.93}$$ Calculate power: $$1.08^{11.93} = e^{11.93 \times \ln(1.08)} = e^{11.93 \times 0.07696} = e^{0.918} = 2.504$$ Population: $$P = 4000 \times 2.504 = 10016$$ (approx) 15. **Answer for Part B:** The populations will be equal at about 10,016 people. **Final answers:** - Part A: Option C - Part B: Approximately 10,016 people