Examkin

Question

Difficulty: MediumDirection and Distance Test

A geologist leaves her base camp and drives a rover 30 km30\text{ km} straight North. From there, she turns 135∘135^{\circ} to her right and travels 102 km10\sqrt{2}\text{ km} in a straight line. She then turns 45∘45^{\circ} to her left and travels 20 km20\text{ km}. Finally, she takes a 90∘90^{\circ} turn to her right and drives another 20 km20\text{ km}. What is her final position relative to the base camp?

  1. A
    30 km30\text{ km} West
  2. B
    10 km10\text{ km} West
  3. 30 km30\text{ km} EastAnswer
  4. D
    30 km30\text{ km} South

Answer

30 km30\text{ km} East
By tracing the movements on a Cartesian plane where North is the positive y-axis and East is the positive x-axis, the sequence unfolds as follows: the initial position is (0,30)(0, 30) after the first drive. A 135∘135^{\circ} right turn points the rover South-East, and moving 102 km10\sqrt{2}\text{ km} adds a displacement of (10,−10)(10, -10), resulting in position (10,20)(10, 20). A 45∘45^{\circ} left turn points the rover East, adding (20,0)(20, 0) to reach (30,20)(30, 20). The final 90∘90^{\circ} right turn points the rover South, adding (0,−20)(0, -20) to land precisely at (30,0)(30, 0). The point (30,0)(30, 0) is exactly 30 km30\text{ km} East of the starting base camp at (0,0)(0, 0).

Step-by-Step Solution

1
Map the first movement from the base camp.
Starting at origin (0,0)(0, 0), moving 30 km30\text{ km} North reaches coordinate (0,30)(0, 30) while facing North.
Establishing the initial position and orientation on a Cartesian plane (where North is +y and East is +x) is necessary for calculating subsequent turns.
2
Calculate the second movement after a 135∘135^{\circ} right turn.
Turning 135∘135^{\circ} right from North changes the heading to South-East. Traveling 102 km10\sqrt{2}\text{ km} South-East yields a displacement of +10+10 in the x-direction and −10-10 in the y-direction. The new position is (10,20)(10, 20).
A 135∘135^{\circ} clockwise turn from North equates to a South-East direction. Using trigonometry, 102×cos⁡(45∘)=1010\sqrt{2} \times \cos(45^{\circ}) = 10 and 102×sin⁡(45∘)=1010\sqrt{2} \times \sin(45^{\circ}) = 10 for the respective axis displacements.
3
Calculate the third movement after a 45∘45^{\circ} left turn.
Turning 45∘45^{\circ} left (counter-clockwise) from South-East aligns the heading to East. Traveling 20 km20\text{ km} East shifts the position by +20+20 on the x-axis, bringing the coordinates to (30,20)(30, 20).
Adjusting the South-East heading by 45∘45^{\circ} back towards the counter-clockwise direction points exactly East.
4
Calculate the final movement after a 90∘90^{\circ} right turn.
Turning 90∘90^{\circ} right (clockwise) from East changes the heading to South. Traveling 20 km20\text{ km} South shifts the position by −20-20 on the y-axis, resulting in a final coordinate of (30,0)(30, 0).
A 90∘90^{\circ} right turn from East is South. Moving 20 km20\text{ km} South precisely negates the remaining +20 km+20\text{ km} vertical displacement.
5
Determine the final relative position.
The final coordinate is (30,0)(30, 0) while the base camp is at (0,0)(0, 0). This represents a displacement of 30 km30\text{ km} directly East.
Comparing the final position coordinates directly to the origin determines the exact distance and direction.

Key Concept

Vector displacement combining angular turns and cardinal directions
Rate this question

Topics

Direction and Distance Test

More questions from this topic