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. 5.20 m
B. 4.33 m
C. 2.60 m
D. 8.66 m

Explanation

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

### Step 1: Understand the Problem
We have a boat man who can row at a speed of 10 km/hr in still water. The river flows at a speed of 5 km/hr, and the width of the river is 2 km. We need to find the time taken to cross the river when the boat man aims to minimize the distance traveled.

### Step 2: Set Up the Velocity Components
- Let \( V_m \) be the speed of the boat man in still water: \( V_m = 10 \) km/hr.
- Let \( V_r \) be the speed of the river: \( V_r = 5 \) km/hr.
- The width of the river (the distance to be crossed) is \( d = 2 \) km.

### Step 3: Determine the Effective Velocity
To minimize the distance traveled, the boat man should row at an angle such that his effective velocity across the river (perpendicular to the flow) is maximized. The effective velocity \( V_{mr} \) of the boat man with respect to the river is given by:

\[ V_{mr} = \sqrt{V_m^2 - V_r^2} \]

### Step 4: Calculate the Effective Velocity
Substituting the values into the equation:

\[ V_{mr} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3} \text{ km/hr} \]

### Step 5: Calculate the Time to Cross the River
The time \( t \) taken to cross the river can be calculated using the formula:

\[ t = \frac{d}{V_{mr}} \]

Substituting the values:

\[ t = \frac{2 \text{ km}}{5\sqrt{3} \text{ km/hr}} = \frac{2}{5\sqrt{3}} \text{ hr} \]

### Step 6: Simplify the Time
To express the time in a more usable form, we can rationalize the denominator:

\[ t = \frac{2 \sqrt{3}}{15} \text{ hr} \]

### Step 7: Convert Time to Minutes
To convert the time from hours to minutes, we multiply by 60:

\[ t = \frac{2 \sqrt{3}}{15} \times 60 = \frac{120 \sqrt{3}}{15} = 8 \sqrt{3} \text{ minutes} \]

### Final Answer
Thus, the time elapsed in rowing the boat is \( 8 \sqrt{3} \) minutes.

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