meg rowed her boat upstream a distance of 91 mi and then rowed back to the starting point. the total time of the trip was 20 hours. if the rate of the current was 3 mph, find the average speed of the boat relative to the water.

meg rowed at an average speed of mph relative to the water.
(simplify your answer.)

Question

meg rowed her boat upstream a distance of 91 mi and then rowed back to the starting point. the total time of the trip was 20 hours. if the rate of the current was 3 mph, find the average speed of the boat relative to the water.

meg rowed at an average speed of mph relative to the water.
(simplify your answer.)

meg rowed at an average speed of   mph relative to the water.
(simplify your answer.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Equations", "Uniform Motion Problems" ], "new_concepts": [], "current_concepts": [ "Rational Equations", "Uniform Motion Problems", "Quadratic Equations" ] } ## Set up the rational equation for total time $$ \begin{aligned} &\text{Let } r \text{ be the average speed of the boat relative to the water (in mph), with } r > 3.\ &\text{Upstream speed: } r - 3\ &\text{Downstream speed: } r + 3\ &\frac{91}{r-3} + \frac{91}{r+3} = 20 \end{aligned} $$ ## Solve the rational equation for r $$ \begin{aligned} &91(r+3) + 91(r-3) = 20(r-3)(r+3)\ &91r + 273 + 91r - 273 = 20(r^2 - 9)\ &182r = 20r^2 - 180\ &20r^2 - 182r - 180 = 0\ &10r^2 - 91r - 90 = 0 \end{aligned} $$ ## Factor and find the positive root $$ \begin{aligned} &(10r + 9)(r - 10) = 0\ &r = 10 \quad (\text{since } r > 3) \end{aligned} $$ 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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QUESTION IMAGE

Question

Explanation:

Set up the rational equation for total time

Solve the rational equation for r

Factor and find the positive root

Answer: