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Zorluk: ZorDirection and Distance Test

A forest conservation officer starting from a watchtower walks 5 km5\text{ km} due North. She then turns 135∘135^\circ clockwise and walks 42 km4\sqrt{2}\text{ km}. Next, she turns 45∘45^\circ counter-clockwise and walks 8 km8\text{ km} straight. Finally, she turns 90∘90^\circ counter-clockwise and walks 4 km4\text{ km} to reach a research station. What is the shortest straight-line distance between the watchtower and the research station, and in which direction is the research station relative to the watchtower?

  1. 13 km13\text{ km}, North-EastCevap
  2. B
    13 km13\text{ km}, South-East
  3. C
    21 km21\text{ km}, North-East
  4. D
    13 km13\text{ km}, South-West

Cevap

The research station is 13 km13\text{ km} away from the watchtower in the North-East direction.
The net horizontal displacement along the East-West axis is 4+8=12 km4 + 8 = 12\text{ km} East, and the net vertical displacement along the North-South axis is 5−4+4=5 km5 - 4 + 4 = 5\text{ km} North. Applying the Pythagorean theorem yields a direct direct distance of 122+52=13 km\sqrt{12^2 + 5^2} = 13\text{ km}. Because both net displacement coordinates are positive, the research station lies to the North-East of the starting watchtower.

Adım Adım Çözüm

1
Set up a Cartesian coordinate system with the starting watchtower at (0,0)(0, 0).
Initial position: (0,0)(0, 0), facing North.
Establishing standard coordinate axes (xx-axis for East/West, yy-axis for North/South) simplifies vector calculation.
2
Calculate position after the first leg (5 km5\text{ km} North).
Position: (0,5)(0, 5), facing North (90∘90^\circ relative to positive xx-axis).
Moving 5 km5\text{ km} North adds 55 units along the positive yy-axis.
3
Calculate position after turning 135∘135^\circ clockwise and walking 42 km4\sqrt{2}\text{ km}.
New orientation is South-East (−45∘-45^\circ). Displacement components are Δx=42cos⁡(−45∘)=4 km\Delta x = 4\sqrt{2}\cos(-45^\circ) = 4\text{ km} East, Δy=42sin⁡(−45∘)=−4 km\Delta y = 4\sqrt{2}\sin(-45^\circ) = -4\text{ km} North (South). Position becomes (0+4,5−4)=(4,1)(0 + 4, 5 - 4) = (4, 1).
Turning 135∘135^\circ clockwise from North points facing to South-East.
4
Calculate position after turning 45∘45^\circ counter-clockwise and walking 8 km8\text{ km}.
New orientation is due East (0∘0^\circ). Displacement Δx=8 km\Delta x = 8\text{ km}, Δy=0\Delta y = 0. Position becomes (4+8,1+0)=(12,1)(4 + 8, 1 + 0) = (12, 1).
Turning 45∘45^\circ counter-clockwise from South-East realigns facing direction to East.
5
Calculate position after turning 90∘90^\circ counter-clockwise and walking 4 km4\text{ km}.
New orientation is due North (90∘90^\circ). Displacement Δx=0\Delta x = 0, Δy=4 km\Delta y = 4\text{ km}. Final position becomes (12,1+4)=(12,5)(12, 1 + 4) = (12, 5).
Turning 90∘90^\circ counter-clockwise from East points facing to North.
6
Compute direct shortest distance and final cardinal orientation relative to origin (0,0)(0, 0).
Shortest distance d=122+52=144+25=169=13 kmd = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\text{ km}. Since both x=12>0x = 12 > 0 and y=5>0y = 5 > 0, the final direction relative to the watchtower is North-East.
The Pythagorean theorem gives straight-line displacement in a 2D Euclidean plane.

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