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`,and the width of the river is `2km`, the time taken to cross the river is (in hours)

Options

A. `2/27`
B. `2/sqrt27`
C. `2/3`
D. `2/sqrt45`

Explanation

To solve the problem of a man swimming across a river with a 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. The river flows at a speed of 3 km/h, and the width of the river is 2 km. We need to find the time taken for the man to swim directly across the river to the point directly opposite his starting point.

### Step 2: Set Up the Components
The swimmer's velocity can be broken down into two components:
- **Vertical Component (Vy)**: This is the component of the swimmer's speed that helps him cross the river.
- **Horizontal Component (Vx)**: This is the component of the swimmer's speed that counters the river's current.

### Step 3: Use the Right Triangle
Using the Pythagorean theorem, we can find the resultant velocity of the swimmer. The swimmer's speed in still water (6 km/h) and the river's speed (3 km/h) form a right triangle:
- Vy = 6 km/h (the swimmer's speed)
- Vx = 3 km/h (the river's speed)

### Step 4: Calculate the Vertical Component
To find the vertical component of the swimmer's velocity (Vy), we can use the sine function:
- \( Vx = 6 \sin(\theta) \)
- \( Vx = 3 \) km/h (to counter the river's flow)

Thus, we have:
\[ 6 \sin(\theta) = 3 \implies \sin(\theta) = \frac{1}{2} \implies \theta = 30^\circ \]

Now, we can find the vertical component:
\[ Vy = 6 \cos(30^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \text{ km/h} \]

### Step 5: Calculate the Time to Cross the River
The time taken to cross the river can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \]
Where:
- Distance = 2 km (width of the river)
- Velocity = \( Vy = 3\sqrt{3} \) km/h

Substituting the values:
\[ \text{Time} = \frac{2 \text{ km}}{3\sqrt{3} \text{ km/h}} = \frac{2}{3\sqrt{3}} \text{ hours} \]

### Final Answer
The time taken to cross the river is:
\[ \text{Time} = \frac{2}{3\sqrt{3}} \text{ hours} \]