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.

Explanation

To solve the problem of the swimmer crossing the river, we can follow these steps:

### Step 1: Understand the Given Information
- Width of the river (D) = 500 m = 0.5 km
- Speed of the river (Vr) = 5 km/h
- Speed of the swimmer with respect to water (Vm) = 3 km/h

### Step 2: Set Up the Coordinate System
- Let the direction across the river (width) be the y-axis.
- Let the direction of the river flow be the x-axis.

### Step 3: Determine the Components of the Swimmer's Velocity
The swimmer's velocity with respect to the water can be broken down into two components:
- \( V_{my} = V_m \cdot \cos(\theta) \) (component across the river)
- \( V_{mx} = V_m \cdot \sin(\theta) \) (component along the river)

Where \( \theta \) is the angle the swimmer makes with the direction perpendicular to the river flow.

### Step 4: Write the Equation for the Time to Cross the River
The time taken to cross the river can be expressed as:
\[ t = \frac{D}{V_{my}} = \frac{0.5 \text{ km}}{V_m \cdot \cos(\theta)} \]

### Step 5: Determine the Condition for Minimum Time
To minimize the time taken to cross the river, we need to maximize \( V_{my} \). Since \( V_{my} = V_m \cdot \cos(\theta) \), the maximum value occurs when \( \cos(\theta) = 1 \) (i.e., when \( \theta = 0^\circ \)). Thus:
\[ V_{my} = V_m = 3 \text{ km/h} \]

### Step 6: Calculate the Minimum Time
Substituting \( V_{my} \) into the time equation:
\[ t = \frac{0.5 \text{ km}}{3 \text{ km/h}} = \frac{1}{6} \text{ hours} \]

### Step 7: Convert Time to Minutes
To convert hours into minutes:
\[ t = \frac{1}{6} \text{ hours} \times 60 \text{ minutes/hour} = 10 \text{ minutes} \]

### Final Answer
The shortest possible time for the swimmer to cross the river is **10 minutes**.

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