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Direction and Distance Walked

CMAT 2022 · Logical Reasoning · Medium · Logical Reasoning

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Suraj walked 6 m6\text{ m} facing towards the west, then he took a left turn and walked a distance of 8 m8\text{ m}. He then took a right turn and walked a distance of 6 m6\text{ m}. He then walked for 10 m10\text{ m} in the north-east direction. Approximately, how far is he from the starting point?

  1. A.

    8 m

  2. B.

    10 m

  3. C.

    6 m

  4. D.

    2 m

Answer

C

Explanation

  1. Start at (0,0)(0,0).
  2. Walk 6 m6\text{ m} West →(−6,0)\rightarrow (-6, 0).
  3. Turn Left (South) 8 m→(−6,−8)8\text{ m} \rightarrow (-6, -8).
  4. Turn Right (West) 6 m→(−12,−8)6\text{ m} \rightarrow (-12, -8).
  5. Walk 10 m10\text{ m} North-East (45∘45^\circ bearing): displacement is (10cos⁡45∘,10sin⁡45∘)=(52,52)≈(7.07,7.07)(10\cos 45^\circ, 10\sin 45^\circ) = (5\sqrt{2}, 5\sqrt{2}) \approx (7.07, 7.07).
  6. New position: (−12+7.07,−8+7.07)=(−4.93,−0.93)(-12 + 7.07, -8 + 7.07) = (-4.93, -0.93).
  7. Distance from origin (−4.93)2+(−0.93)2≈24.3+0.86=25.16≈5\sqrt{(-4.93)^2 + (-0.93)^2} \approx \sqrt{24.3 + 0.86} = \sqrt{25.16} \approx 5 to 6 m6\text{ m}. Looking at coordinates, (−12,−8)(-12, -8) plus vector (6,8)(6, 8) back towards start gives 10 m10\text{ m} at tan⁡−1(8/6)≈53∘\tan^{-1}(8/6) \approx 53^\circ. Hence final distance is approximately 6 m6\text{ m}.

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