A boat which has a speed of 5 kilometre per hour in still water crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in kilometre per hour is:

To find the velocity of the river water, we need to analyze the situation step by step. The boat crosses a river that is 1 kilometer wide in 15 minutes, and we know its speed in still water is 5 kilometers per hour. Let's break this down to find the speed of the river current.

Understanding the Problem

The boat's speed in still water is 5 km/h, which means if there were no current, it could travel 5 kilometers in one hour. However, the river current affects the boat's actual path and speed across the river.

Calculating the Boat's Effective Speed

First, we need to determine how far the boat travels in 15 minutes. Since 15 minutes is a quarter of an hour, we can calculate the distance the boat would cover in that time:

• Distance = Speed × Time
• Distance = 5 km/h × 0.25 h = 1.25 km

This means that the boat travels a total distance of 1.25 kilometers in 15 minutes. However, it only needs to cross a width of 1 kilometer directly across the river.

Finding the River's Current Velocity

To cross the river along the shortest path, the boat must have a component of its velocity that directly counters the river's current. We can visualize this using a right triangle where:

• The width of the river (1 km) is one leg of the triangle.
• The distance traveled by the boat (1.25 km) is the hypotenuse.
• The other leg represents the downstream distance caused by the current.

Using the Pythagorean theorem, we can express this relationship:

• (Distance across the river)² + (Distance downstream)² = (Total distance traveled)²
• (1 km)² + (x)² = (1.25 km)²

Calculating this gives:

• 1 + x² = 1.5625
• x² = 0.5625
• x = √0.5625 = 0.75 km

Calculating the Velocity of the River

The distance downstream (0.75 km) is traveled in the same 15 minutes. To find the river's velocity, we convert this distance into speed:

• Time taken = 15 minutes = 0.25 hours
• Velocity of the river = Distance downstream / Time = 0.75 km / 0.25 h = 3 km/h

Final Result

Thus, the velocity of the river water is 3 kilometers per hour. This means that while the boat is trying to go straight across, the current pushes it downstream, and it has to compensate for that to reach the opposite bank directly.