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 (in hours) and the horizontal distance travelled (in km) are respectively

Options

A. `3/4,9/4`
B. `3/5,3`
C. `1/4,15/4`
D. `3/sqrt7,7`

Explanation

To solve the problem of a man swimming across a river, we need to determine the time taken to cross the river and the horizontal distance traveled. Let's break it down step by step.

### Step 1: Understand the Problem
The man swims in still water at a speed of 4 km/h, while the river flows at a speed of 3 km/h. He wants to cross the river in the shortest time possible. The width of the river is given as 3 km.

### Step 2: Determine the Effective Speed
To cross the river in the shortest time, the man should swim directly across the river (perpendicular to the flow). His effective speed across the river is his swimming speed, which is 4 km/h.

### 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 is the width of the river (3 km) and the speed is the man's swimming speed (4 km/h).

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

### Step 4: Calculate the Horizontal Distance Traveled
While the man swims directly across, the river's current will carry him downstream. To find the horizontal distance traveled, we need to calculate how far the river carries him during the time he takes to cross.

The horizontal distance can be calculated using the river's speed:
\[ \text{Horizontal Distance} = \text{Speed of River} \times \text{Time} \]
The speed of the river is 3 km/h, and we already calculated the time as \( \frac{3}{4} \) hours.

\[ \text{Horizontal Distance} = 3 \text{ km/h} \times \frac{3}{4} \text{ hours} = \frac{9}{4} \text{ km} = 2.25 \text{ km} \]

### Final Answers
- Time taken to cross the river: \( \frac{3}{4} \) hours
- Horizontal distance traveled: \( 2.25 \) km

### Summary
Thus, the time taken to cross the river is \( \frac{3}{4} \) hours, and the horizontal distance traveled is \( 2.25 \) km.

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