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. `12.8kmh^(-1)`
B. `6kmh^(-1)`
C. `8kmh^(-1)`
D. `10kmh^(-1)`

Answer

To solve the problem, we need to find the velocity of the river water given the velocity of the boat and the resultant velocity when crossing the river.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Velocity of the boat (Vb) = 8 km/h
- Resultant velocity (V) = 10 km/h

2. **Understand the Problem:**
- The boat is moving across the river, and the river has its own velocity (Vr).
- The boat's velocity and the river's velocity are perpendicular to each other, forming a right triangle with the resultant velocity.

3. **Use the Pythagorean Theorem:**
- According to the Pythagorean theorem, we can relate the velocities as follows:
\[ V^2 = Vb^2 + Vr^2 \]
- Here, V is the resultant velocity, Vb is the velocity of the boat, and Vr is the velocity of the river.

4. **Substitute the Known Values:**
- Substitute V = 10 km/h and Vb = 8 km/h into the equation:
\[ (10)^2 = (8)^2 + Vr^2 \]
- This simplifies to:
\[ 100 = 64 + Vr^2 \]

5. **Solve for Vr:**
- Rearranging the equation gives:
\[ Vr^2 = 100 - 64 \]
\[ Vr^2 = 36 \]
- Taking the square root of both sides:
\[ Vr = \sqrt{36} = 6 \text{ km/h} \]

6. **Conclusion:**
- The velocity of the river water (Vr) is 6 km/h.

### Final Answer:
The velocity of the river water is **6 km/h**.

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