Difficulty: Medium
Correct Answer: 4 km
Explanation:
Introduction / Context:
This question is from the direction sense test topic and involves a person moving in several different directions, including diagonal directions such as south east and north west. The aim is to determine the straight line distance from the starting point, which is Laxmi's office, after she completes all the movements. Problems like this check understanding of basic coordinate ideas and vector addition without requiring heavy mathematics.
Given Data / Assumptions:
Concept / Approach:
The core concept is to resolve each movement into horizontal (east west) and vertical (north south) components and then add them. South east and north west are at 45 degrees, so the horizontal and vertical components of those displacements are equal in magnitude. When two diagonal movements of the same length occur in opposite diagonal directions, many components cancel out, which often simplifies the calculation. We then combine all east west movements and all north south movements to get the net displacement from the office.
Step-by-Step Solution:
Step 1: Place the office at the origin (0, 0) of a coordinate system with east positive x and north positive y.Step 2: A 6 km south east move has components 6 / √2 km east and 6 / √2 km south. Call 6 / √2 = a.Step 3: After this move, Laxmi is at (a, −a).Step 4: She then walks 15 km west, which subtracts 15 from the x coordinate, giving (a − 15, −a).Step 5: The next 6 km north west move has components a west and a north, so adding (−a, a) gives the new position (a − 15 − a, −a + a) = (−15, 0).Step 6: Finally, she walks 11 km east, which adds 11 to the x coordinate, placing her at (−15 + 11, 0) = (−4, 0).Step 7: The point (−4, 0) is 4 km to the west of the office, so the straight line distance from the office is 4 km.
Verification / Alternative check:
An intuitive check is to notice that the two 6 km diagonal moves cancel each other in both north south and east west components. The south east move takes her some amount east and the same amount south, while the north west move brings her back by the same amount west and north. This leaves only the pure horizontal movements of 15 km west and 11 km east, which combine to a net 4 km towards the west. Since no north south displacement remains, the distance is exactly 4 km, matching the coordinate calculation.
Why Other Options Are Wrong:
Options 10 km and 11 km would require a much larger net displacement, which is inconsistent because most of the long legs cancel each other. The option 6 km might be chosen by simply subtracting 15 and 11 without noticing the diagonal cancellation, or by guessing based on one of the step lengths. The option 8 km is another rough guess that does not correspond to any correct Pythagorean combination arising from the data. Only 4 km matches the precise net horizontal displacement from the office.
Common Pitfalls:
A common mistake is to try to treat diagonal distances as if they were purely horizontal or purely vertical, which leads to incorrect totals. Another frequent error is to add distances algebraically without considering direction, for example adding 6 + 15 + 6 + 11 and then trying to guess the result. Not drawing even a simple rough diagram makes it easy to lose track of the cancelling movements. Thinking in terms of components along east west and north south axes helps keep the logic clear and avoids arithmetic confusion.
Final Answer:
After completing all the given movements, Laxmi is located 4 km away from her office, towards the west, so the required distance is 4 km.
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