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Difficulty: MediumDirection and Distance Test

An autonomous delivery drone is programmed to navigate from a central warehouse to a remote drop-off location. The drone executes the following sequence of movements:

1. It takes off and flies 10 km10\text{ km} due North.
2. It turns 90∘90^\circ to its right and flies 20 km20\text{ km}.
3. It then rotates 135∘135^\circ anti-clockwise and flies 102 km10\sqrt{2}\text{ km}.
4. Finally, it turns 135∘135^\circ clockwise and flies 5 km5\text{ km} to reach the destination.

What is the shortest direct distance between the warehouse and the drop-off destination?

  1. A
    15 km15\text{ km}
  2. B
    20 km20\text{ km}
  3. C
    35 km35\text{ km}
  4. 25 km25\text{ km}Answer

Answer

The shortest direct distance between the warehouse and the drop-off destination is 25 km25\text{ km}.
The correct answer is found by tracking the drone's position on a 2D plane. By applying the vector components of each movement, the net displacement is 15 km15\text{ km} East and 20 km20\text{ km} North. The shortest direct distance is the hypotenuse of these components: 152+202=25 km\sqrt{15^2 + 20^2} = 25\text{ km}.

Step-by-Step Solution

1
Determine the drone's position after the first two movements.
Starting at origin (0,0)(0,0), the drone flies 10 km10\text{ km} North to (0,10)(0, 10). Turning 90∘90^\circ right faces it East; flying 20 km20\text{ km} brings it to (20,10)(20, 10).
Tracking coordinates on a plane allows for precise calculation of vector displacements.
2
Calculate the coordinates after the 135∘135^\circ anti-clockwise rotation.
Facing East, a 135∘135^\circ anti-clockwise turn points the drone North-West. A distance of 102 km10\sqrt{2}\text{ km} NW means moving 10 km10\text{ km} West and 10 km10\text{ km} North. The new position is (20−10,10+10)=(10,20)(20-10, 10+10) = (10, 20).
The North-West vector components of 10210\sqrt{2} are split equally into (−10,+10)(-10, +10) on a standard coordinate grid.
3
Calculate the final position after the 135∘135^\circ clockwise rotation.
Facing North-West, a 135∘135^\circ clockwise turn points the drone East. Flying 5 km5\text{ km} East adds +5+5 to the x-coordinate. The final position is (10+5,20)=(15,20)(10+5, 20) = (15, 20).
Establishing the final drop-off location coordinates relative to the warehouse to allow for distance calculation.
4
Compute the shortest direct distance using the final coordinates.
Using the Pythagorean theorem: Distance =152+202=225+400=625=25 km= \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25\text{ km}.
The shortest direct distance corresponds to the hypotenuse of the right-angled triangle formed by the net horizontal and vertical displacements.

Key Concept

Vector displacement, angular rotation tracking, and Pythagorean theorem application.
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Topics

Direction and Distance Test

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