A river 5 Km wide is flowing at the rate of 2 Km/H. The minimum time taken by a boat to cross the river with a speed 5 Km/H (in still water) is approximately

Options

A. 150 min
B. 30 min
C. 60 min
D. 90 min

Answer

To solve the problem of how long it takes for a boat to cross a river that is 5 km wide while the river flows at 2 km/h and the boat moves at 5 km/h in still water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- The river is 5 km wide.
- The speed of the river current is 2 km/h.
- The speed of the boat in still water is 5 km/h.

2. **Determine the Effective Speed of the Boat**:
- The boat's speed relative to the river is 5 km/h. However, to cross the river directly, the boat must aim upstream to counteract the current.
- The effective speed of the boat perpendicular to the river (across the river) is given by the formula:
\[ V_{\text{effective}} = \sqrt{V_{\text{boat}}^2 - V_{\text{river}}^2} \]
- Here, \( V_{\text{boat}} = 5 \text{ km/h} \) and \( V_{\text{river}} = 2 \text{ km/h} \).

3. **Calculate the Effective Speed**:
- Substitute the values into the formula:
\[ V_{\text{effective}} = \sqrt{5^2 - 2^2} = \sqrt{25 - 4} = \sqrt{21} \text{ km/h} \]
- Calculate \( \sqrt{21} \):
\[ V_{\text{effective}} \approx 4.58 \text{ km/h} \]

4. **Calculate the Time to Cross the River**:
- The time taken to cross the river can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
- Here, the distance is the width of the river (5 km) and the speed is the effective speed of the boat (approximately 4.58 km/h):
\[ \text{Time} = \frac{5 \text{ km}}{4.58 \text{ km/h}} \approx 1.09 \text{ hours} \]

5. **Convert Time to Minutes**:
- To convert hours into minutes, multiply by 60:
\[ \text{Time in minutes} \approx 1.09 \times 60 \approx 65.4 \text{ minutes} \]

### Final Answer:
The minimum time taken by the boat to cross the river is approximately **65.4 minutes**.