A snail boat sails 2 km due East, 5 km `37^(@)` South of East and finally an unknown displacement. The final displacement of the boat from the starting point is 6 km due East and, the third displacement is x km due North. Find x

Explanation

To solve the problem, we need to analyze the displacements of the snail boat step by step.

### Step 1: Define the Displacements
1. The first displacement is 2 km due East.
- In vector form, this can be represented as:
\[ \mathbf{S_1} = 2 \hat{i} \]

2. The second displacement is 5 km at an angle of \(37^\circ\) South of East.
- We can break this displacement into its components:
- The x-component (East direction) is:
\[ S_{2x} = 5 \cos(37^\circ) \]
- The y-component (South direction) is:
\[ S_{2y} = -5 \sin(37^\circ) \quad (\text{negative because it's South}) \]
- Therefore, the second displacement vector is:
\[ \mathbf{S_2} = 5 \cos(37^\circ) \hat{i} - 5 \sin(37^\circ) \hat{j} \]

### Step 2: Calculate the Components of the Second Displacement
Using the approximate values of \(\cos(37^\circ) \approx 0.8\) and \(\sin(37^\circ) \approx 0.6\):
- The x-component becomes:
\[ S_{2x} = 5 \times 0.8 = 4 \text{ km} \]
- The y-component becomes:
\[ S_{2y} = -5 \times 0.6 = -3 \text{ km} \]

Thus, the second displacement vector can be expressed as:
\[ \mathbf{S_2} = 4 \hat{i} - 3 \hat{j} \]

### Step 3: Define the Third Displacement
Let the third displacement be \(x\) km due North. In vector form, this is:
\[ \mathbf{S_3} = 0 \hat{i} + x \hat{j} \]

### Step 4: Sum of Displacements
The net displacement from the starting point is given as 6 km due East. In vector form, this is:
\[ \mathbf{S_{net}} = 6 \hat{i} \]

Now, we can express the net displacement as the sum of the three displacements:
\[ \mathbf{S_{net}} = \mathbf{S_1} + \mathbf{S_2} + \mathbf{S_3} \]

Substituting the values we have:
\[ 6 \hat{i} = (2 \hat{i}) + (4 \hat{i} - 3 \hat{j}) + (0 \hat{i} + x \hat{j}) \]

### Step 5: Combine the Components
Combining the i and j components:
- For the i components:
\[ 6 = 2 + 4 + 0 \implies 6 = 6 \quad (\text{This is satisfied}) \]
- For the j components:
\[ 0 = -3 + x \]

### Step 6: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = 3 \]

### Conclusion
The value of \(x\) is 3 km. Therefore, the third displacement is 3 km due North.