A swimmer wishes to cross a `500 - m` river flowing at `5 km h^-1`. His speed with respect to water is `3 km h^-1`. The shortest possible time to cross the river is.

Options

A. `10 min`
B. `20 min`
C. `6 min`
D. `7.5 min`

Explanation

To solve the problem of the swimmer crossing a 500 m wide river flowing at 5 km/h with a swimming speed of 3 km/h relative to the water, we can follow these steps:

### Step 1: Convert Units
First, we need to convert all the speeds into the same unit. The width of the river is given in meters, and we will convert the speeds from km/h to m/s.

- Speed of the river (V_r) = 5 km/h = (5 * 1000 m) / (3600 s) = 1.39 m/s
- Speed of the swimmer with respect to water (V_s) = 3 km/h = (3 * 1000 m) / (3600 s) = 0.83 m/s

### Step 2: Analyze the Motion
The swimmer's effective speed across the river depends on the angle at which he swims. To cross the river in the shortest time, he should swim directly across, which means he will have to compensate for the river's current.

### Step 3: Calculate the Time to Cross
The time taken to cross the river can be calculated using the formula:
\[ t = \frac{d}{v_{\text{effective}}} \]
where \( d \) is the width of the river (500 m) and \( v_{\text{effective}} \) is the component of the swimmer's speed that is directed across the river.

Since the swimmer swims directly across the river, his effective speed across the river is equal to his speed with respect to water, which is 0.83 m/s.

### Step 4: Substitute Values
Now we can substitute the values into the formula:
\[ t = \frac{500 \text{ m}}{0.83 \text{ m/s}} \]

### Step 5: Calculate the Time
Calculating this gives:
\[ t \approx 602.41 \text{ seconds} \]

### Step 6: Convert Time to Minutes
To convert seconds into minutes:
\[ t \approx \frac{602.41}{60} \approx 10.04 \text{ minutes} \]

### Final Answer
Thus, the shortest possible time to cross the river is approximately **10 minutes**.

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