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 find the velocity of the river stream given the width of the river, the speed of the boat, and the time taken to cross the river. Let's break this down step by step.

### Step-by-Step Solution:

1. **Understand the Problem**:
- The width of the river (d) = 1 km.
- The speed of the boat (v_b) = 5 km/hr.
- The time taken to cross the river (t) = 15 minutes = 15/60 hours = 1/4 hours.

2. **Calculate the Distance Covered by the Boat**:
- The boat covers the width of the river in the shortest path, which is directly across the river.
- Distance covered (d) = width of the river = 1 km.

3. **Use the Formula for Speed**:
- Speed (v) = Distance / Time.
- Here, we can rearrange the formula to find the resultant speed of the boat in the direction of the river's flow.
- We know that the time taken to cross the river is 1/4 hours, so:
\[ v_{resultant} = \frac{d}{t} = \frac{1 \text{ km}}{\frac{1}{4} \text{ hr}} = 4 \text{ km/hr} \]

4. **Set Up the Relationship Between Velocities**:
- The resultant velocity (v_r) of the boat is the vector sum of the boat's velocity (v_b) and the river's velocity (v_river).
- Since the boat is moving perpendicular to the river's flow, we can use the Pythagorean theorem:
\[ v_{resultant}^2 = v_b^2 - v_{river}^2 \]
- Substituting the known values:
\[ 4^2 = 5^2 - v_{river}^2 \]
\[ 16 = 25 - v_{river}^2 \]

5. **Solve for the Velocity of the River**:
- Rearranging gives:
\[ v_{river}^2 = 25 - 16 = 9 \]
- Taking the square root:
\[ v_{river} = 3 \text{ km/hr} \]

### Final Answer:
The velocity of the river stream is **3 km/hr**.