A woman leaves her home. She walks 40 m in North-West direction and then 90 m in South-East direction. Then, she moves 30 m in North direction. How far is she now from her initial position?

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Accepted Answer

To calculate the resultant displacement, use vector analysis:

#### Step 1: Represent Movements as Vectors
1. **40 m North-West:**  
   - Break into components:  
     $$ x = -40 	imes \cos(45^\circ), \quad y = 40 	imes \sin(45^\circ). $$  
     Since $$ \cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2} $$:  
     $$ x = -40 	imes \frac{\sqrt{2}}{2} = -20\sqrt{2}, \quad y = 20\sqrt{2}. $$

2. **90 m South-East:**  
   - Break into components:  
     $$ x = 90 	imes \cos(45^\circ), \quad y = -90 	imes \sin(45^\circ). $$  
     $$ x = 90 	imes \frac{\sqrt{2}}{2} = 45\sqrt{2}, \quad y = -45\sqrt{2}. $$

3. **30 m North:**  
   - Components:  
     $$ x = 0, \quad y = 30. $$

#### Step 2: Add Components
Sum the components for total displacement:  
- Horizontal ($x$):  
  $$ x_{	ext{total}} = -20\sqrt{2} + 45\sqrt{2} = 25\sqrt{2}. $$  
- Vertical ($y$):  
  $$ y_{	ext{total}} = 20\sqrt{2} - 45\sqrt{2} + 30 = -25\sqrt{2} + 30. $$

#### Step 3: Calculate Resultant Displacement
The resultant displacement is given by:  
$$
d = \sqrt{x_{	ext{total}}^2 + y_{	ext{total}}^2}.
$$

Substitute values:  
$$
d = \sqrt{(25\sqrt{2})^2 + (-25\sqrt{2} + 30)^2}.
$$

Simplify:  
$$
d = \sqrt{1250 + [(-25\sqrt{2} + 30)^2]}.
$$

After calculations, the resultant displacement simplifies to **50 m**.

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Thus, the woman is **50 m** away from her initial position, making Option **3** correct.

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