A man who can swin with a velocity `v_(b)` in still water wants to cross a river a river in minimum possible time, if the river is flowing with a velocity `v_(w)` and b , x are the width and drift respectively, the actual distance travelled by the man is .

Options

A. x
B. b
C. `sqrt(b^(2)+x^(2))`
D. `sqrt(b^(2)-x^(2))`

Answer

To solve the problem, we need to determine the actual distance the man travels while swimming across the river, taking into account the drift caused by the river's current.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The man swims with a velocity \( v_b \) in still water.
- The river flows with a velocity \( v_w \).
- The width of the river is \( b \).
- The drift caused by the river current is \( x \).

2. **Swimming at an Angle**:
- To minimize the time taken to cross the river, the man should swim at an angle \( \theta \) relative to the flow of the river. This angle allows him to counteract the current while still making progress across the river.

3. **Identifying the Distances**:
- When the man swims at angle \( \theta \), he will reach a point downstream due to the river's current. The distance directly across the river is \( b \), and the distance downstream (drift) is \( x \).

4. **Using the Pythagorean Theorem**:
- The actual distance \( s \) that the man travels can be visualized as the hypotenuse of a right triangle where one leg is the width of the river \( b \) and the other leg is the drift \( x \).
- According to the Pythagorean theorem, the relationship between the sides of the triangle is given by:
\[ s = \sqrt{b^2 + x^2} \]

5. **Conclusion**:
- Therefore, the actual distance traveled by the man while crossing the river is:
\[ s = \sqrt{b^2 + x^2} \]

### Final Answer:
The actual distance traveled by the man is \( \sqrt{b^2 + x^2} \).