A boat which has a speed of 6km/h in still water crosses a river of width 1 km along the shortest possible path in 20 min. the velocity of the river water in km/h is

Options

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

Explanation

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

### Step-by-Step Solution:

1. **Convert Time to Hours**:
The time taken to cross the river is given as 20 minutes. We need to convert this into hours for consistency with the speed units (km/h).
\[ \text{Time} = \frac{20 \text{ minutes}}{60} = \frac{1}{3} \text{ hours} \]

**Hint**: Remember to convert minutes to hours when dealing with speed in km/h.

2. **Identify Given Values**:
- Speed of the boat in still water, \( V_b = 6 \) km/h
- Width of the river, \( d = 1 \) km
- Time taken to cross the river, \( t = \frac{1}{3} \) hours

3. **Calculate Effective Velocity Across the River**:
The boat crosses the river in the shortest path, which means it is moving perpendicular to the current of the river. The effective velocity of the boat across the river can be calculated using the formula:
\[ \text{Effective Velocity} = \frac{\text{Distance}}{\text{Time}} = \frac{1 \text{ km}}{\frac{1}{3} \text{ hours}} = 3 \text{ km/h} \]

**Hint**: Use the formula for speed, which is distance divided by time.

4. **Apply Pythagorean Theorem**:
The velocity of the boat with respect to the ground can be represented as the resultant of the boat's speed in still water and the river's speed. According to the Pythagorean theorem:
\[ V_g^2 = V_b^2 - V_r^2 \]
where \( V_g \) is the effective velocity across the river, \( V_b \) is the speed of the boat in still water, and \( V_r \) is the velocity of the river.

5. **Substituting Known Values**:
Substitute the known values into the equation:
\[ (3)^2 = (6)^2 - V_r^2 \]
\[ 9 = 36 - V_r^2 \]

**Hint**: Make sure to square the velocities correctly.

6. **Rearranging the Equation**:
Rearranging the equation to solve for \( V_r^2 \):
\[ V_r^2 = 36 - 9 = 27 \]

7. **Calculating the Velocity of the River**:
Taking the square root to find \( V_r \):
\[ V_r = \sqrt{27} = 3\sqrt{3} \text{ km/h} \]

**Hint**: When taking the square root, remember that \( \sqrt{27} \) can be simplified to \( 3\sqrt{3} \).

8. **Final Answer**:
The velocity of the river water is:
\[ V_r \approx 5.196 \text{ km/h} \quad (\text{approximately } 5.2 \text{ km/h}) \]

### Summary:
The velocity of the river water is approximately \( 3\sqrt{3} \) km/h or about 5.2 km/h.