1. **State the problem:** A tour boat travels 9 miles upstream and 9 miles downstream in a total of 4 hours. The river current speed is 3 miles per hour. We need to find the speed of the boat in still water. 2. **Define variables:** Let $b$ be the speed of the boat in still water (in miles per hour). 3. **Speeds upstream and downstream:** - Upstream speed = $b - 3$ - Downstream speed = $b + 3$ 4. **Time formula:** Time = Distance / Speed. - Time upstream = $\frac{9}{b-3}$ - Time downstream = $\frac{9}{b+3}$ 5. **Total time given:** $$\frac{9}{b-3} + \frac{9}{b+3} = 4$$ 6. **Solve the equation:** Multiply both sides by $(b-3)(b+3)$ to clear denominators: $$9(b+3) + 9(b-3) = 4(b-3)(b+3)$$ 7. **Simplify left side:** $$9b + 27 + 9b - 27 = 4(b^2 - 9)$$ $$18b = 4b^2 - 36$$ 8. **Rearrange to standard quadratic form:** $$4b^2 - 18b - 36 = 0$$ 9. **Divide entire equation by 2 to simplify:** $$\cancel{2} \times (2b^2 - 9b - 18) = 0$$ $$2b^2 - 9b - 18 = 0$$ 10. **Use quadratic formula:** $$b = \frac{9 \pm \sqrt{(-9)^2 - 4 \times 2 \times (-18)}}{2 \times 2} = \frac{9 \pm \sqrt{81 + 144}}{4} = \frac{9 \pm \sqrt{225}}{4}$$ 11. **Calculate roots:** $$b = \frac{9 \pm 15}{4}$$ 12. **Possible solutions:** - $$b = \frac{9 + 15}{4} = \frac{24}{4} = 6$$ - $$b = \frac{9 - 15}{4} = \frac{-6}{4} = -1.5$$ (not physically possible since speed cannot be negative) 13. **Final answer:** The speed of the boat in still water is $6$ miles per hour.