A river is flowing from west to east at a speed of 5 metres per minute.A man on the south bank of the river, capable of swimming at 10 metres per minute in still water, wants to swim across the river in the shortest time. He should swim in a direction.

Options

A. due north
B. `30^0` east of north
C. `30^0` north of west
D. `60^0` east of north

Answer

To solve the problem of how the man should swim across the river in the shortest time, we can break down the solution step by step.

### Step-by-Step Solution:

1. **Identify the Speeds**:
- The speed of the river (Vr) is 5 meters per minute (from west to east).
- The swimming speed of the man in still water (Vm) is 10 meters per minute.

2. **Understanding the Objective**:
- The man wants to swim across the river (from the south bank to the north bank) in the shortest time possible.

3. **Determine the Direction of Swimming**:
- To minimize the time taken to cross the river, we need to analyze the angle at which the man should swim.
- If the man swims directly across (perpendicular to the river flow), he will be swept downstream by the current.

4. **Using Trigonometry**:
- Let’s denote the angle at which the man swims relative to the north direction as θ.
- The component of the man's swimming speed that is directed across the river (perpendicular to the flow) is given by:
\[ V_{m \text{ across}} = V_m \cdot \cos(\theta) \]
- The component of the man's swimming speed that is directed along the river (parallel to the flow) is:
\[ V_{m \text{ along}} = V_m \cdot \sin(\theta) \]

5. **Setting Up the Time Equation**:
- The time taken to cross the river (width d) can be expressed as:
\[ t = \frac{d}{V_{m \text{ across}}} = \frac{d}{V_m \cdot \cos(\theta)} \]
- Since the river is flowing, the man will also drift downstream while swimming. The effective speed across the river must be maximized to minimize time.

6. **Maximizing the Cosine Component**:
- To minimize time, we need to maximize the component \( V_{m \text{ across}} \).
- The cosine function \( \cos(\theta) \) reaches its maximum value when \( \theta = 0^\circ \). This means the man should swim directly north.

7. **Conclusion**:
- Therefore, to cross the river in the shortest time, the man should swim directly north (at an angle of 0 degrees to the north).

### Final Answer:
The man should swim in the direction **north** to cross the river in the shortest time.