A boat man can row with a speed of 10 km/hr. in still water. The river flow steadily at 5 km/hr. and the width of the river is 2 km. if the boat man cross the river with reference to minimum distance of approach then time elapsed in rowing the boat will be:-

Options

A. `(2sqrt(3))/(5)` hour
B. `(2)/(5)sqrt(3)` hour
C. `(3sqrt(2))/(5)`hour
D. `(5sqrt(2))/(3)` hour

Answer

To solve the problem step by step, we need to analyze the motion of the boatman in the river.

### Step 1: Understand the Problem
The boatman rows across a river that is 2 km wide, with a rowing speed of 10 km/hr in still water and a river current of 5 km/hr. We need to find the time taken to cross the river when he aims for the minimum distance of approach.

### Step 2: Determine the Components of Velocity
The boatman's velocity can be broken down into two components:
- The velocity perpendicular to the river (across the river): \( V_b = 10 \, \text{km/hr} \)
- The velocity of the river (downstream): \( V_r = 5 \, \text{km/hr} \)

### Step 3: Calculate the Resultant Velocity
To find the resultant velocity of the boatman with respect to the ground, we use the Pythagorean theorem:
\[ V_{resultant} = \sqrt{V_b^2 - V_r^2} \]
Substituting the values:
\[ V_{resultant} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3} \, \text{km/hr} \]

### Step 4: Calculate the Time to Cross the River
The time taken to cross the river can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} \]
Here, the distance to cross is the width of the river (2 km), and the velocity is the resultant velocity calculated above:
\[ \text{Time} = \frac{2 \, \text{km}}{5\sqrt{3} \, \text{km/hr}} = \frac{2}{5\sqrt{3}} \, \text{hours} \]

### Final Answer
Thus, the time elapsed in rowing the boat is:
\[ \frac{2}{5\sqrt{3}} \, \text{hours} \]