40 m
50 m
To calculate the woman's final distance from her starting position, we will analyze her movements using vector displacements.
Step 1: Her first movement is 40 m North-West. This can be resolved into:
Step 2: Her second movement is 90 m South-East. This can be resolved into:
Step 3: Her third movement is 30 m North. We now sum the north-south displacements:
\(\frac{40}{\sqrt{2}} + 30 - \frac{90}{\sqrt{2}}\)
= \( \frac{40\sqrt{2} + 30\sqrt{2} - 90\sqrt{2}}{2} \)
= \( \frac{-20\sqrt{2}}{2} \)
= \(-10\sqrt{2}\) m South
For the 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
Her net displacement is \(-10\sqrt{2}\) m South and \(25\sqrt{2}\) m East from her starting point. The resultant distance from the starting point is calculated as:
\(\sqrt{(-10\sqrt{2})^2 + (25\sqrt{2})^2}\)
= \(\sqrt{200 + 1250}\)
= \(\sqrt{1450}\)
= \( \sqrt{25 \times 58} \)
= \(5\sqrt{58}\approx 40.63\) m
Therefore, the closest value among the options is 40 m.