A hard question on boats and currents for P6, with a step-by-step solution, diagrams and the answer. Boats going with and against the current appear in the Hua Xia Cup's P5 and P6 first rounds, qualifying rounds and finals.
The question
A cruise boat is heading upstream.
In still water it goes 70 m a minute, and the river flows at 15 m a minute; both speeds stay the same.
On the way, a lifebuoy falls off the stern and drifts downstream with the water.
3 minutes later the crew notice; the boat turns at once and goes back downstream after it (ignore the time taken to turn).
When the boat reaches the lifebuoy, how far is the lifebuoy from where it fell in?
Show the hints (3)
💡Hint 1: In the 3 minutes after it falls, which way do the boat and the lifebuoy each go, and how fast do they separate?
💡Hint 2: Once the boat turns, how much more does it cover than the lifebuoy each minute?
💡Hint 3: They are 210 m apart; the boat closes 70 m a minute. For how many minutes in all does the lifebuoy drift?
Worked solution
Have a go first, then read the solution step by step.
1.In 3 minutes the boat goes (70 − 15) × 3 = 165 m upstream and the lifebuoy 15 × 3 = 45 m downstream: 165 + 45 = 210 m apart.
2.Chasing, the boat does 70 + 15 = 85 m a minute and the lifebuoy 15, so the gap closes by 85 − 15 = 70 m a minute.
3.210 ÷ 70 = 3 minutes to catch it.
4.The lifebuoy drifts for 3 + 3 = 6 minutes in all: 15 × 6 = 90 m.
⚠️Common mistake: counting only the 3 minutes before it was noticed and answering 45 m. The lifebuoy keeps drifting during the chase.
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