A river is flowing from west to east with a speed of `5 m//min.` A man can swim in still water with a velocity `10m//min.` In which direction should the man swim so as to take the shortest possible path to go to the south.

Options

A. `30^(@)` with downstream
B. `60^(@)` with downstream
C. `120^(@)` with downstream
D. South

Explanation

To solve the problem of determining the direction in which a man should swim to take the shortest possible path to go south while a river flows from west to east, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the velocities**:
- The river flows from west to east with a velocity \( U = 5 \, \text{m/min} \).
- The man can swim in still water with a velocity \( V = 10 \, \text{m/min} \).

2. **Understand the goal**:
- The man wants to swim straight south. However, due to the river's current, he must swim at an angle upstream to counteract the eastward flow of the river.

3. **Set up the vector diagram**:
- Let the direction south be represented as the negative y-axis and the direction of the river (east) as the positive x-axis.
- The man's swimming direction will be at an angle \( \theta \) from the south towards the west.

4. **Use the Pythagorean theorem**:
- The man's swimming velocity can be broken down into two components:
- The southward component: \( V \sin(\theta) \)
- The westward component: \( V \cos(\theta) \)
- To ensure he swims directly south, the westward component must equal the river's velocity:
\[ V \cos(\theta) = U \]
- Substituting the known values:
\[ 10 \cos(\theta) = 5 \]
- Solving for \( \cos(\theta) \):
\[ \cos(\theta) = \frac{5}{10} = \frac{1}{2} \]

5. **Calculate the angle**:
- Using the inverse cosine function:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

6. **Determine the final direction**:
- Since the angle \( \theta \) is measured from the south towards the west, the man should swim at an angle of \( 60^\circ \) from the south towards the west.
- To express this in terms of the downstream direction (east), we can say:
\[ 90^\circ + 60^\circ = 150^\circ \]
- Thus, the man should swim at an angle of \( 150^\circ \) from the east (downstream).

### Final Answer:
The man should swim at an angle of \( 60^\circ \) from the south towards the west, or equivalently, \( 150^\circ \) from the east (downstream).