A boat crosses a river of width 200m in the shortest time and is found to experience a drift to 100m in reaching the opposite bank. The time taken now is 't' . If the same boat is to cross the river by shortest path. The time taken to cross will be

Options

A. 2t
B. `sqrt(2)t`
C. 3t
D. `(2t)/(sqrt(3))`

Answer

To solve the problem step by step, we will analyze the motion of the boat across the river, considering the drift and the time taken for both the shortest time and the shortest path.

### Step 1: Understand the Given Information
- Width of the river (W) = 200 m
- Drift experienced by the boat = 100 m
- Time taken to cross the river in the shortest time = t

### Step 2: Analyze the Shortest Time Crossing
When the boat crosses the river in the shortest time, it is directed straight across the river, which means it is not compensating for the current. The boat's speed (V) can be related to the width of the river and the time taken:
\[ V \sin(\theta) = \frac{W}{t} \]
Since \(\theta = 90^\circ\) for the shortest time, we have:
\[ V = \frac{200}{t} \]

### Step 3: Analyze the Drift
The drift experienced by the boat is due to the current of the river. The drift can be expressed as:
\[ U t = 100 \]
Where U is the speed of the river. From this, we can find U:
\[ U = \frac{100}{t} \]

### Step 4: Set Up for Shortest Path Crossing
For the shortest path across the river, the boat must travel at an angle (α) to compensate for the current. The effective speed of the boat across the river is given by:
\[ V_{\text{effective}} = \sqrt{V^2 - U^2} \]

### Step 5: Find the Time for Shortest Path Crossing
The time taken to cross the river by the shortest path (T) can be expressed as:
\[ T = \frac{W}{V_{\text{effective}}} \]
Substituting for \(V_{\text{effective}}\):
\[ T = \frac{200}{\sqrt{V^2 - U^2}} \]

### Step 6: Substitute Values of V and U
Now we substitute the values of V and U obtained earlier:
\[ T = \frac{200}{\sqrt{\left(\frac{200}{t}\right)^2 - \left(\frac{100}{t}\right)^2}} \]
\[ T = \frac{200}{\sqrt{\frac{40000}{t^2} - \frac{10000}{t^2}}} \]
\[ T = \frac{200}{\sqrt{\frac{30000}{t^2}}} \]
\[ T = \frac{200}{\frac{100\sqrt{3}}{t}} \]
\[ T = \frac{200t}{100\sqrt{3}} \]
\[ T = \frac{2t}{\sqrt{3}} \]

### Final Answer
Thus, the time taken to cross the river by the shortest path is:
\[ T = \frac{2t}{\sqrt{3}} \]