A boat moves with speed of 5 km/h relative to water in a river flowing with a speed of 3 km/h and having a width of 1 km. The minimum time taken around a round trip is

Options

A. 5 min
B. 60 min
C. 20 min
D. 30 min

Explanation

To solve the problem of the boat's round trip across a river, we can break down the solution into clear steps:

### Step 1: Understand the Problem
We have a boat moving with a speed of 5 km/h relative to the water. The river flows with a speed of 3 km/h, and the width of the river is 1 km. We need to find the minimum time taken for a round trip across the river.

### Step 2: Determine the Effective Speed of the Boat
The boat's speed relative to the ground when moving across the river can be calculated using the Pythagorean theorem. The effective speed of the boat \( v \) when moving perpendicular to the current can be found as follows:

\[ v_{\text{effective}} = \sqrt{(v_{\text{boat}})^2 - (v_{\text{river}})^2} \]

Where:
- \( v_{\text{boat}} = 5 \) km/h (speed of the boat relative to water)
- \( v_{\text{river}} = 3 \) km/h (speed of the river)

### Step 3: Calculate the Effective Speed
Substituting the values into the equation:

\[ v_{\text{effective}} = \sqrt{(5)^2 - (3)^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ km/h} \]

### Step 4: Calculate the Time to Cross the River
The time taken to cross the river (one way) can be calculated using the formula:

\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

The distance to cross the river is 1 km, and the effective speed is 4 km/h:

\[ \text{Time to cross one way} = \frac{1 \text{ km}}{4 \text{ km/h}} = 0.25 \text{ hours} \]

### Step 5: Calculate the Total Time for the Round Trip
Since the round trip consists of going to the other bank and returning, the total time is:

\[ \text{Total Time} = 2 \times \text{Time to cross one way} = 2 \times 0.25 \text{ hours} = 0.5 \text{ hours} \]

### Step 6: Convert Time to Minutes
To convert the time from hours to minutes:

\[ 0.5 \text{ hours} = 0.5 \times 60 \text{ minutes} = 30 \text{ minutes} \]

### Final Answer
The minimum time taken for the round trip is **30 minutes**.

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