A man can swim in still water at a speed of 6 kmph and he has to cross the river and reach just opposite point on the other bank. If the river is flowing at a speed of 3 kmph, he has to swim in the direction

Options

A. 30° with the river flow
B. 60° with river flow
C. 135° with the river flow
D. 120° with the river flow

Answer

To solve the problem of the man swimming across the river while accounting for the current, we can break it down into a series of steps:

### Step 1: Understand the Problem
The man swims in still water at a speed of 6 km/h, and the river flows at a speed of 3 km/h. He needs to swim in such a way that he reaches a point directly opposite his starting point on the other bank.

### Step 2: Set Up the Vectors
We can represent the velocities as vectors:
- Let \( V_m \) be the velocity of the man in still water (6 km/h).
- Let \( V_r \) be the velocity of the river (3 km/h).
- The resultant velocity \( V_{mr} \) should point directly across the river (from point A to point B).

### Step 3: Analyze the Angles
Let \( \theta \) be the angle between the man's swimming direction and the direction directly across the river. The angle between the man's swimming direction and the river's current will be \( 90^\circ + \theta \).

### Step 4: Use Trigonometry
Using the sine rule for the triangle formed by the velocities:
\[ \sin(\theta) = \frac{V_r}{V_m} \]
Substituting the values:
\[ \sin(\theta) = \frac{3}{6} = \frac{1}{2} \]

### Step 5: Calculate \( \theta \)
From the sine function:
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \]

### Step 6: Determine the Angle \( \alpha \)
Since \( \alpha = 90^\circ + \theta \):
\[ \alpha = 90^\circ + 30^\circ = 120^\circ \]

### Step 7: Conclusion
The man must swim at an angle of \( 120^\circ \) with respect to the direction of the river flow to reach the point directly opposite on the other bank.

### Final Answer
The angle at which the man has to swim is \( 120^\circ \) against the river flow.

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