A boatman can row with a speed of `10 kmh^(-1)` in still water. River flows at `6 km h^(-1)` . If he crosses the river from one bank to the other along the shortest possible path, time taken to cross that river of width 1 km is

Options

A. `1/8 h`
B. `1/4h`
C. `1/2h`
D. `1 h`

Explanation

To solve the problem of how long it takes for the boatman to cross the river, we can follow these steps:

### Step 1: Understand the Given Information
- Speed of the boat in still water (Vb) = 10 km/h
- Speed of the river (Vr) = 6 km/h
- Width of the river (D) = 1 km

### Step 2: Determine the Effective Speed of the Boat
To cross the river in the shortest path, the boatman must row at an angle against the current of the river. The effective speed of the boat across the river can be calculated using the Pythagorean theorem, where the boat's speed in the direction perpendicular to the river's flow is given by:

\[ V_{\text{effective}} = V_b \cdot \cos(\theta) \]

### Step 3: Calculate the Angle
To ensure that the boat crosses directly across the river, the component of the boat's speed in the direction of the river's flow must equal the speed of the river:

\[ V_b \cdot \sin(\theta) = V_r \]

Substituting the values:

\[ 10 \cdot \sin(\theta) = 6 \]

From this, we can find:

\[ \sin(\theta) = \frac{6}{10} = 0.6 \]

### Step 4: Calculate Cosine of the Angle
Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):

\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - (0.6)^2 = 1 - 0.36 = 0.64 \]

Thus:

\[ \cos(\theta) = \sqrt{0.64} = 0.8 \]

### Step 5: Calculate the Effective Speed
Now we can calculate the effective speed of the boat across the river:

\[ V_{\text{effective}} = V_b \cdot \cos(\theta) = 10 \cdot 0.8 = 8 \text{ km/h} \]

### Step 6: Calculate the Time to Cross the River
The time taken to cross the river can be calculated using the formula:

\[ t = \frac{D}{V_{\text{effective}}} \]

Substituting the values:

\[ t = \frac{1 \text{ km}}{8 \text{ km/h}} = \frac{1}{8} \text{ hours} \]

### Step 7: Convert Time to Minutes
To convert hours into minutes:

\[ t = \frac{1}{8} \text{ hours} \times 60 \text{ minutes/hour} = 7.5 \text{ minutes} \]

### Final Answer
The time taken to cross the river is \( \frac{1}{8} \) hours or 7.5 minutes.

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