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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Uniform Motion Problems", "Rational Equations", "Quadratic Formula" ], "new_concepts": [], "current_concepts": [ "Uniform Motion Problems", "Rational Equations", "Quadratic Equations" ] } ## Set up the rational equation for total trip time $$ \begin{aligned} &\text{Let } r \text{ be the average speed of the boat relative to the water (in mph), where } r > 9.\ &\text{Upstream speed: } r - 9\ &\text{Downstream speed: } r + 9\ &\text{Total time equation: } \frac{19}{r-9} + \frac{19}{r+9} = 20 \end{aligned} $$ ## Solve the rational equation for r $$ \begin{aligned} &19(r+9) + 19(r-9) = 20(r-9)(r+9)\ &19r + 171 + 19r - 171 = 20(r^2 - 81)\ &38r = 20r^2 - 1620\ &20r^2 - 38r - 1620 = 0\ &10r^2 - 19r - 810 = 0 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} &r = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(10)(-810)}}{2(10)}\ &r = \frac{19 \pm \sqrt{361 + 32400}}{20}\ &r = \frac{19 \pm \sqrt{32761}}{20}\ &r = \frac{19 \pm 181}{20}\ &r = \frac{200}{20} = 10 \quad (\text{since } r > 9) \end{aligned} $$ Meg rowed at an average speed of 10 mph relative to the water. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Rational Equations" ] }

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Set up the rational equation for total trip time

Solve the rational equation for r

Solve the quadratic equation

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