A 400 m wide river is flowing from west to east at velocity 3m/s. A boat starts from one bank and it crosses the river along the shortest path. When it reaches the middle of river, wind starts blowing horizontally along north-east at a velocity `sqrt(2)` m/s . Due to this, the direction of rowing of boat has to be changed for the boat to move along the shortest path. If the velocity of boat in still water with no wind blowing is 5 m/s, then choose the correct option(s).

Options

A. The initial direction of rowing of boat is at an angle of` 53^(@)` with upstream direction
B. Time taken to reach the middle of river is 50 seconds
C. After the wind starts blowing, the direction of rowing of boat is `37^(@)` with upstream direction
D. Total time taken by the boat to cross the river along the shortest path is 100 seconds

Explanation

To solve the problem step by step, we will analyze the motion of the boat across the river, taking into account the river's current and the wind's influence.

### Step 1: Analyze the Initial Conditions
- The river is 400 m wide and flows from west to east at a velocity of 3 m/s.
- The boat has a velocity of 5 m/s in still water.
- The boat aims to cross the river along the shortest path, which means it must move straight across (perpendicular to the banks).

### Step 2: Determine the Initial Rowing Angle
To ensure the boat crosses directly across the river, we need to find the angle at which the boat should row upstream to counteract the river's current.

Using the equation for the x-component of the boat's velocity:
\[ V_{BR} \cos(\theta_1) + V_{river} = 0 \]
Where:
- \( V_{BR} = 5 \, \text{m/s} \) (velocity of the boat),
- \( V_{river} = 3 \, \text{m/s} \).

Setting the x-component to zero:
\[ 5 \cos(\theta_1) + 3 = 0 \]
\[ 5 \cos(\theta_1) = -3 \]
\[ \cos(\theta_1) = \frac{3}{5} \]

Thus,
\[ \theta_1 = \cos^{-1}\left(\frac{3}{5}\right) \approx 53^\circ \]

### Step 3: Calculate Time to Reach the Middle of the River
The distance to the middle of the river is 200 m. The y-component of the boat's velocity is:
\[ V_{BY} = V_{BR} \sin(\theta_1) = 5 \sin(53^\circ) \approx 5 \cdot \frac{4}{5} = 4 \, \text{m/s} \]

Now, we can find the time taken to reach the middle of the river:
\[ t_1 = \frac{200 \, \text{m}}{4 \, \text{m/s}} = 50 \, \text{s} \]

### Step 4: Analyze the Effect of Wind
At the middle of the river, wind starts blowing at a velocity of \( \sqrt{2} \, \text{m/s} \) towards the northeast (45 degrees).

### Step 5: Determine the New Rowing Angle
Now, we need to find the new angle \( \theta_2 \) that the boat must take to continue moving straight across the river.

The x-component of the net velocity must still be zero:
\[ V_{BR} \cos(\theta_2) + V_{river} + V_{wind} \cos(45^\circ) = 0 \]
Where:
- \( V_{wind} = \sqrt{2} \, \text{m/s} \).

Substituting the values:
\[ 5 \cos(\theta_2) + 3 + \frac{\sqrt{2}}{2} = 0 \]
\[ 5 \cos(\theta_2) = -3 - \frac{\sqrt{2}}{2} \]

Calculating \( \frac{\sqrt{2}}{2} \approx 0.707 \):
\[ 5 \cos(\theta_2) = -3 - 0.707 = -3.707 \]
\[ \cos(\theta_2) = \frac{-3.707}{5} \approx -0.7414 \]

Thus,
\[ \theta_2 = \cos^{-1}\left(-0.7414\right) \approx 37^\circ \]

### Step 6: Calculate Time for the Second Half of the River
The distance for the second half of the river is again 200 m. The y-component of the boat's velocity is now:
\[ V_{BY} = V_{BR} \sin(\theta_2) = 5 \sin(37^\circ) \approx 5 \cdot \frac{3}{5} = 3 \, \text{m/s} \]

Now, we can find the time taken to cross the second half:
\[ t_2 = \frac{200 \, \text{m}}{3 \, \text{m/s}} \approx 66.67 \, \text{s} \]

### Step 7: Calculate Total Time Taken
The total time taken to cross the river is:
\[ T = t_1 + t_2 = 50 \, \text{s} + 66.67 \, \text{s} \approx 116.67 \, \text{s} \]

### Final Answer
The total time taken by the boat to cross the river in the shortest path is approximately 116.67 seconds.