A boat crosses a river with a velocity of `8(km)/(h)`. If the resulting velocity of boat is `10(km)/(h)` then the velocity of river water is

Options

A. `4(km)/(h)`
B. `6(km)/(h)`
C. `8(km)/(h)`
D. `10(km)/(h)`

Explanation

To solve the problem, we can use the concept of vector addition. The boat's velocity and the river's velocity can be represented as vectors, and the resultant velocity can be found using the Pythagorean theorem since the two velocities are perpendicular to each other.

### Step-by-Step Solution:

1. **Identify the velocities**:
- Let the velocity of the boat relative to the ground (resultant velocity) be \( V_r = 10 \, \text{km/h} \).
- Let the velocity of the boat in still water be \( V_b = 8 \, \text{km/h} \).
- Let the velocity of the river water be \( V_w \) (which we need to find).

2. **Set up the equation using the Pythagorean theorem**:
Since the boat's velocity and the river's velocity are perpendicular to each other, we can use the Pythagorean theorem:
\[ V_r^2 = V_b^2 + V_w^2 \]

3. **Substitute the known values**:
Substitute \( V_r = 10 \, \text{km/h} \) and \( V_b = 8 \, \text{km/h} \) into the equation:
\[ (10)^2 = (8)^2 + V_w^2 \]
This simplifies to:
\[ 100 = 64 + V_w^2 \]

4. **Solve for \( V_w^2 \)**:
Rearranging the equation gives:
\[ V_w^2 = 100 - 64 \]
\[ V_w^2 = 36 \]

5. **Take the square root to find \( V_w \)**:
\[ V_w = \sqrt{36} = 6 \, \text{km/h} \]

6. **Conclusion**:
The velocity of the river water is \( 6 \, \text{km/h} \).

### Final Answer:
The velocity of the river water is \( 6 \, \text{km/h} \).