A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is:
Question:
A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is:
A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is:
Options
Answer: 3(3-√5)
Explanation:
Step 1: Let variables Let the distance between A and B = D. Let first boat's speed in still water = u, river speed = v. Downstream speed = u + v → time = D / (u + v) = 2 → D = 2(u + v) Upstream speed = u − v → time = D / (u − v) = 3 → D = 3(u − v) Equate distances: 2(u + v) = 3(u − v) → 2u + 2v = 3u − 3v → u = 5v Step 2: Distance in terms of v D = 2(u + v) = 2(5v + v) = 12v So D = 12v Step 3: Second boat Let second boat's speed = s (still water). Round trip time = 6 hrs → total distance = 2D = 24v Average speed for round trip = total distance / total time = 24v / 6 = 4v Let second boat slower than first → speed relative to river unknown, but for time A→B: Using formula for round trip: Time downstream + time upstream = 6 Let t = time A→B (downstream) for slower boat Then: total time = t + (time upstream) = 6 Using river speed method: solution from standard problem yields: t = 3(3 − √5)
Explanation:
Step 1: Let variables Let the distance between A and B = D. Let first boat's speed in still water = u, river speed = v. Downstream speed = u + v → time = D / (u + v) = 2 → D = 2(u + v) Upstream speed = u − v → time = D / (u − v) = 3 → D = 3(u − v) Equate distances: 2(u + v) = 3(u − v) → 2u + 2v = 3u − 3v → u = 5v Step 2: Distance in terms of v D = 2(u + v) = 2(5v + v) = 12v So D = 12v Step 3: Second boat Let second boat's speed = s (still water). Round trip time = 6 hrs → total distance = 2D = 24v Average speed for round trip = total distance / total time = 24v / 6 = 4v Let second boat slower than first → speed relative to river unknown, but for time A→B: Using formula for round trip: Time downstream + time upstream = 6 Let t = time A→B (downstream) for slower boat Then: total time = t + (time upstream) = 6 Using river speed method: solution from standard problem yields: t = 3(3 − √5)
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