The width of river is 1 km. The velocity of boat is 5 km/hr. The boat covered the width of river with shortest will possible path in 15 min. Then the velocity of river stream is:

Options

A. 3 km/hr
B. 4 km/hr
C. `sqrt(29)` km/hr
D. `sqrt(41)` km/hr

Explanation

To solve the problem, we need to determine the velocity of the river stream based on the given information about the boat and the river. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a river that is 1 km wide, a boat that travels at a velocity of 5 km/hr, and the boat crosses the river in 15 minutes. We need to find the velocity of the river stream.

### Step 2: Convert Time to Hours
Since the velocity is given in km/hr, we need to convert the time taken from minutes to hours.
- 15 minutes = 15/60 hours = 1/4 hours.

### Step 3: Calculate the Resultant Velocity
The boat crosses the river in the shortest path, which means the resultant velocity of the boat (V) must be perpendicular to the banks of the river. The width of the river (D) is 1 km.

Using the formula for speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
We can calculate the resultant velocity (V):
\[ V = \frac{1 \text{ km}}{1/4 \text{ hr}} = 4 \text{ km/hr} \]

### Step 4: Relate the Velocities
According to the Pythagorean theorem, the relationship between the boat's velocity (B), the river's velocity (R), and the resultant velocity (V) can be expressed as:
\[ V^2 = B^2 - R^2 \]
Where:
- \( B = 5 \text{ km/hr} \) (velocity of the boat)
- \( V = 4 \text{ km/hr} \) (resultant velocity)
- \( R \) is the velocity of the river stream.

### Step 5: Substitute and Solve for R
Substituting the known values into the equation:
\[ 4^2 = 5^2 - R^2 \]
\[ 16 = 25 - R^2 \]
Rearranging gives:
\[ R^2 = 25 - 16 \]
\[ R^2 = 9 \]

### Step 6: Calculate R
Taking the square root of both sides:
\[ R = \sqrt{9} = 3 \text{ km/hr} \]

### Conclusion
The velocity of the river stream is **3 km/hr**.

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