Question

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?

• A. 30 m
• B. 60 m
• C. 40 m
• D. 50 m

Answer 1

The Correct Option is C

To determine how far the woman is from her initial position, we need to analyze her path step by step using vector displacements. 

Step 1: She first walks 40 m in the North-West direction. In terms of vector components, this movement can be split into:

• Northward: \( \frac{40}{\sqrt{2}} \) m
• Westward: \( \frac{40}{\sqrt{2}} \) m

Step 2: Next, she walks 90 m in the South-East direction. This can be divided into:

• Southward: \( \frac{90}{\sqrt{2}} \) m
• Eastward: \( \frac{90}{\sqrt{2}} \) m

Step 3: She further moves 30 m North. Adding up all north-south movements, we calculate her net north-south displacement:

\(\frac{40}{\sqrt{2}} + 30 - \frac{90}{\sqrt{2}}\)

= \( \frac{40\sqrt{2} + 30\sqrt{2} - 90\sqrt{2}}{2} \)

= \( \frac{-20\sqrt{2}}{2} \)

= \(-10\sqrt{2}\) m South

For east-west movements:

\(-\frac{40}{\sqrt{2}} + \frac{90}{\sqrt{2}}\)

= \( \frac{90\sqrt{2} - 40\sqrt{2}}{2} \)

= \( \frac{50\sqrt{2}}{2} \)

= 25\sqrt{2}\) m East

Now, she's \(-10\sqrt{2}\) m South and \(25\sqrt{2}\) m East from her starting point. Her resultant distance from the starting point is given by:

\(\sqrt{(-10\sqrt{2})^2 + (25\sqrt{2})^2}\)

= \(\sqrt{200 + 1250}\)

= \(\sqrt{1450}\)

= \( \sqrt{25 \times 58} \)

= \(5\sqrt{58}\approx 40.63\) m

Thus, given the options, the closest value is 40 m, which is the correct answer.