1. **State the problem:** Find the values of $a$ and $b$ for the exponential function $y = ab^x$ passing through points $(3, 20)$ and $(5, 4.05)$. 2. **Write the system of equations:** Using the points, we have: $$20 = ab^3$$ $$4.05 = ab^5$$ 3. **Divide the second equation by the first to eliminate $a$:** $$\frac{4.05}{20} = \frac{ab^5}{ab^3} = b^{5-3} = b^2$$ 4. **Simplify the fraction:** $$\frac{4.05}{20} = 0.2025$$ So, $$b^2 = 0.2025$$ 5. **Solve for $b$ by taking the square root:** $$b = \sqrt{0.2025} = 0.45$$ 6. **Substitute $b=0.45$ back into the first equation to find $a$:** $$20 = a(0.45)^3$$ Calculate $0.45^3$: $$0.45^3 = 0.45 \times 0.45 \times 0.45 = 0.091125$$ So, $$20 = a \times 0.091125$$ 7. **Solve for $a$:** $$a = \frac{20}{0.091125}$$ Show cancellation: $$a = \frac{20}{\cancel{0.091125}} \times \frac{1}{\cancel{0.091125}} = 219.6$$ --- 8. **State the problem:** Write an equation modeling the population $y$ of a town starting at 14,500 in 2010, increasing by 3.5% each year. 9. **Use the exponential growth formula:** $$y = a(1 + r)^t$$ where $a$ is initial population, $r$ is growth rate, and $t$ is years after 2010. 10. **Substitute values:** $$a = 14500, \quad r = 0.035$$ So, $$y = 14500(1.035)^t$$ **Final answers:** - For question 3: $$a \approx 219.6, \quad b = 0.45$$ - For question 4: $$y = 14500(1.035)^t$$