man wants to swim across a river of which 200 m along the shortest path . If the speed of river stream is `3 km h^(-1)` and speed of swimmer in still water is `5 km h^(-1)` , then the time of crossing the river is

Options

A. 10 min
B. 15 min
C. 3 min
D. 6 min

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the problem
We need to find the time taken by a man to swim across a river of width 200 meters while considering the speed of the river and the swimmer.

### Step 2: Convert units
Convert the width of the river from meters to kilometers:
- Width of the river = 200 m = \( \frac{200}{1000} = \frac{1}{5} \) km.

### Step 3: Identify the speeds
- Speed of the river stream, \( V_R = 3 \) km/h.
- Speed of the swimmer in still water, \( V_{MR} = 5 \) km/h.

### Step 4: Determine the effective speed of the swimmer
To swim across the river in the shortest path, the swimmer must swim at an angle against the current. The effective speed of the swimmer across the river can be calculated using the Pythagorean theorem:
\[ V_M = \sqrt{V_{MR}^2 - V_R^2} \]
Substituting the values:
\[ V_M = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ km/h} \]

### Step 5: Calculate the time taken to cross the river
Using the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):
\[ \text{Time} = \frac{\frac{1}{5} \text{ km}}{4 \text{ km/h}} = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20} \text{ hours} \]

### Step 6: Convert time to minutes
To convert hours into minutes, multiply by 60:
\[ \text{Time in minutes} = \frac{1}{20} \times 60 = 3 \text{ minutes} \]

### Final Answer
The time taken to cross the river is **3 minutes**.

---