A man can swim in still water at a speed of 4 kmph. He desires to cross a river flowing at a speed of 3 kmph in the shortest time interval. If the width of the river is 3km time taken to cross the river (in hours) and the horizontal distance travelled (in km) are respectively

Options

A. `1 ms^(-1)`
B. `0.25 ms^(-1)`
C. `0.67 ms^(-1)`
D. `3 ms^(-1)`

Answer

To solve the problem, we need to determine the time taken for the man to cross the river and the horizontal distance he travels while crossing. We can break down the solution into several steps.

### Step 1: Understand the Problem
The man swims in still water at a speed of 4 km/h, and the river flows at a speed of 3 km/h. The width of the river is 3 km. We need to find the time taken to cross the river and the horizontal distance traveled.

### Step 2: Determine the Effective Velocity
To cross the river in the shortest time, the man should swim directly across the river. His swimming velocity will be perpendicular to the flow of the river.

- The man's swimming speed (Vm) = 4 km/h (perpendicular to the river)
- The river's speed (Vr) = 3 km/h (parallel to the river)

### Step 3: 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 to be crossed (width of the river) is 3 km, and the effective speed in the direction of crossing (y-direction) is the man's swimming speed (4 km/h).

\[ \text{Time} = \frac{3 \text{ km}}{4 \text{ km/h}} = 0.75 \text{ hours} \]

### Step 4: Calculate the Horizontal Distance Traveled
While the man swims across the river, he is also carried downstream by the river's current. The horizontal distance (x) he travels due to the river's flow can be calculated using the river's speed and the time taken to cross.

\[ \text{Horizontal Distance} = \text{Speed of River} \times \text{Time} \]
\[ \text{Horizontal Distance} = 3 \text{ km/h} \times 0.75 \text{ hours} = 2.25 \text{ km} \]

### Final Results
- Time taken to cross the river = 0.75 hours
- Horizontal distance traveled = 2.25 km

### Summary
The time taken to cross the river is 0.75 hours, and the horizontal distance traveled is 2.25 km.

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