A person reaches on a point directly opposite on the other bank of a river.The velocity of the water in the river is `4 m//s`and the velocity of the person in still water is `5 m//s`.If the width of the river is `84.6m`, time taken to cross the river in seconds is

Options

A. `28.2`
B. `9.4`
C. 2
D. `84.6`

Explanation

To solve the problem of a person crossing a river, we will follow these steps:

### Step 1: Understand the Problem
We need to find the time taken for a person to cross a river of width 84.6 m, given that the velocity of the river is 4 m/s and the velocity of the person in still water is 5 m/s. The person reaches a point directly opposite on the other bank.

### Step 2: Identify the Relevant Velocities
- Velocity of the river (\(v_r\)) = 4 m/s (this is the horizontal component).
- Velocity of the person in still water (\(v_p\)) = 5 m/s (this is the resultant velocity).

### Step 3: Determine the Effective Velocity Across the River
Since the person must reach directly opposite, we need to find the component of the person's velocity that is perpendicular to the river bank. This can be calculated using the Pythagorean theorem.

Let \(v_{p_y}\) be the effective velocity of the person across the river:
\[ v_{p_y} = \sqrt{v_p^2 - v_r^2} \]
Substituting the values:
\[ v_{p_y} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3 \text{ m/s} \]

### Step 4: Calculate the Time Taken to Cross the River
Using the formula for time, which is:
\[ t = \frac{\text{Distance}}{\text{Velocity}} \]
Here, the distance is the width of the river (84.6 m) and the effective velocity is \(v_{p_y}\) (3 m/s):
\[ t = \frac{84.6 \text{ m}}{3 \text{ m/s}} = 28.2 \text{ seconds} \]

### Conclusion
The time taken for the person to cross the river is approximately **28.2 seconds**.

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