man wants to swim across a river of which 200 m along the shortest path . If the speed of river stream is `3 km h^(-1)` and speed of swimmer in still water is `5 km h^(-1)` , then the time of crossing the river is

Options

A. 10 min
B. 15 min
C. 3 min
D. 6 min

Answer

To solve the problem of a man swimming across a river, we need to determine the time it takes for him to cross a distance of 200 meters, given the speed of the river and the speed of the swimmer in still water. Here’s a step-by-step solution:

### Step 1: Understand the Given Data
- Distance to swim across the river (D) = 200 meters
- Speed of the river (Vr) = 3 km/h
- Speed of the swimmer in still water (Vs) = 5 km/h

### Step 2: Convert Speeds to Consistent Units
Since the distance is given in meters, we should convert the speeds from km/h to m/s:
- To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

\[ Vr = 3 \text{ km/h} = 3 \times \frac{5}{18} \text{ m/s} = \frac{15}{18} \text{ m/s} = \frac{5}{6} \text{ m/s} \approx 0.833 \text{ m/s} \]

\[ Vs = 5 \text{ km/h} = 5 \times \frac{5}{18} \text{ m/s} = \frac{25}{18} \text{ m/s} \approx 1.389 \text{ m/s} \]

### Step 3: Determine the Effective Velocity of the Swimmer
The swimmer's effective velocity (Vs') in the direction across the river can be calculated using the Pythagorean theorem, since the swimmer's velocity and the river's velocity are perpendicular to each other.

\[ Vs' = \sqrt{Vs^2 - Vr^2} \]

Substituting the values we converted:

\[ Vs' = \sqrt{\left(\frac{25}{18}\right)^2 - \left(\frac{5}{6}\right)^2} \]

Calculating each term:

\[ Vs'^2 = \left(\frac{25}{18}\right)^2 - \left(\frac{5}{6}\right)^2 = \frac{625}{324} - \frac{25}{36} \]

To subtract these fractions, we need a common denominator. The common denominator of 324 and 36 is 324.

\[ \frac{25}{36} = \frac{25 \times 9}{36 \times 9} = \frac{225}{324} \]

Now we can perform the subtraction:

\[ Vs'^2 = \frac{625}{324} - \frac{225}{324} = \frac{400}{324} = \frac{100}{81} \]

Taking the square root:

\[ Vs' = \sqrt{\frac{100}{81}} = \frac{10}{9} \text{ m/s} \approx 1.111 \text{ m/s} \]

### Step 4: Calculate the Time to Cross the River
Now, we can find the time (t) taken to cross the river using the formula:

\[ t = \frac{D}{Vs'} \]

Substituting the values:

\[ t = \frac{200 \text{ m}}{\frac{10}{9} \text{ m/s}} = 200 \times \frac{9}{10} = 180 \text{ seconds} \]

### Step 5: Convert Time to Minutes
To convert seconds into minutes:

\[ t = \frac{180 \text{ seconds}}{60} = 3 \text{ minutes} \]

### Final Answer
The time taken for the man to swim across the river is **3 minutes**.

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