A boat which has a speed of `5 km// hr ` in steel water crosses a river of width ` 1 km` along the shortest possible path in `15 minutes`. The velocity of the river water in ` km// hr` is

Options

A. 1
B. 3
C. 4
D. `sqrt(41)`

Explanation

To solve the problem, we need to find the velocity of the river water given the speed of the boat in still water and the time taken to cross the river.

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Speed of the boat in still water, \( V_m = 5 \, \text{km/hr} \)
- Width of the river, \( d = 1 \, \text{km} \)
- Time taken to cross the river, \( t = 15 \, \text{minutes} = \frac{15}{60} \, \text{hr} = \frac{1}{4} \, \text{hr} \)

2. **Calculate the Speed of the Boat Across the River**:
- The speed at which the boat crosses the river can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{d}{t} = \frac{1 \, \text{km}}{\frac{1}{4} \, \text{hr}} = 4 \, \text{km/hr} \]
- Thus, the effective speed of the boat across the river, \( V_{mr} = 4 \, \text{km/hr} \).

3. **Use Pythagorean Theorem**:
- Since the boat's velocity in still water and the river's velocity form a right triangle with the effective velocity across the river, we can use the Pythagorean theorem:
\[ V_m^2 = V_{mr}^2 + V_r^2 \]
- Here, \( V_r \) is the velocity of the river.

4. **Substitute the Known Values**:
- Substitute \( V_m = 5 \, \text{km/hr} \) and \( V_{mr} = 4 \, \text{km/hr} \):
\[ 5^2 = 4^2 + V_r^2 \]
\[ 25 = 16 + V_r^2 \]

5. **Solve for the Velocity of the River**:
- Rearranging gives:
\[ V_r^2 = 25 - 16 = 9 \]
\[ V_r = \sqrt{9} = 3 \, \text{km/hr} \]

6. **Final Answer**:
- The velocity of the river water is \( 3 \, \text{km/hr} \).