To solve this problem, we need to analyze the motion of the boat in relation to the river's current. We have two key pieces of information: the speed of the boat relative to the water and the speed of the current. Let's break down the situation step by step.
Understanding the Boat's Motion
The boat's speed in still water is 6 km/h, and the current flows at 3 km/h. When the boat is crossing the river, it has to contend with the current, which affects its actual path and speed across the river.
Crossing the River
When the boat crosses from point A to point B, it takes 10 minutes. To find the width of the river, we first convert the time into hours:
- 10 minutes = 10/60 hours = 1/6 hours
During this time, the boat is moving across the river while also being pushed downstream by the current. The effective speed of the boat across the river can be calculated using the Pythagorean theorem, as the boat's speed and the current form a right triangle.
Calculating Effective Speed
The effective speed of the boat across the river (perpendicular to the current) can be determined as follows:
- Let the width of the river be W (in km).
- The time taken to cross is 1/6 hours.
- Using the formula: Speed = Distance / Time, we have:
Effective speed across the river = W / (1/6) = 6W km/h.
Since the boat's speed relative to the water is 6 km/h, we can set up the equation:
- 6W = 6 (the speed of the boat in still water).
Thus, W = 1 km. This means the width of the river is 1 km.
Return Journey Analysis
Now, let's analyze the return journey from point B back to point A, which takes 20 minutes (or 1/3 hours). During this time, the boat is also affected by the current, which pushes it downstream.
Effective Speed on Return
On the return trip, the effective speed of the boat can be calculated as follows:
- The downstream speed of the boat is the speed of the boat in still water minus the speed of the current:
- Effective speed downstream = 6 km/h - 3 km/h = 3 km/h.
Now, we can calculate how far downstream the boat is pushed during the return journey:
- Distance downstream = Speed × Time = 3 km/h × (1/3) hours = 1 km.
Summary of Findings
In summary, we have determined the following:
- The width of the river is 1 km.
- The downstream distance from point A to point B is also 1 km.
This analysis shows how the current affects the boat's path and the time taken to cross the river, illustrating the interplay between speed, distance, and time in a real-world scenario.
