A surveyor mapping a nature reserve starts from a base station. He walks 14\text{ km} due … — State PSC Exam

A surveyor mapping a nature reserve starts from a base station. He walks $14\text{ km}$ due East, turns $90^\circ$ to his right, and walks $20\text{ km}$ . Next, he turns $135^\circ$ to his left and walks $10\sqrt{2}\text{ km}$ . Finally, he turns $45^\circ$ to his left and walks $3\text{ km}$ due North to reach an observation point. What is the shortest straight-line distance (in km) between the base station and the observation point?

Answer: 25 km

Answer

The shortest straight-line distance between the base station and the observation point is 25 km.

Resolving each displacement into Cartesian coordinates gives a net horizontal displacement of

\Delta x = 14 + 10 = 24\text{ km}

(East) and a net vertical displacement of

\Delta y = -20 + 10 + 3 = -7\text{ km}

(South). Applying the Pythagorean theorem yields

\sqrt{24^2 + (-7)^2} = \sqrt{625} = 25\text{ km}

Step-by-Step Solution

Establish a Cartesian coordinate system with the base station as the origin

(0,0)

Initial coordinates are

(0,0)

facing East.

Assigning coordinates simplifies multi-turn vector displacement tracking.

Calculate coordinates after moving

14\text{ km}

East, turning

90^\circ

right (facing South), and walking

20\text{ km}

Position is

(14, -20)

facing South.

Moving East adds

14

to the x-coordinate, and moving South subtracts

20

from the y-coordinate.

Resolve the

10\sqrt{2}\text{ km}

walk after a

135^\circ

left turn from South into components.

Facing North-East, displacement components are

+10\text{ km}

East and

+10\text{ km}

North, yielding coordinates

(24, -10)

Turning

135^\circ

left from South points directly North-East (

45^\circ

north of east). The components are

10\sqrt{2}\cos(45^\circ) = 10\text{ km}

East and

10\sqrt{2}\sin(45^\circ) = 10\text{ km}

North.

Update coordinates after turning

45^\circ

left from North-East (facing North) and walking

3\text{ km}

Final position is

(24, -10 + 3) = (24, -7)

45^\circ

left turn aligns the direction due North, adding

3\text{ km}

to the y-coordinate.

Calculate the straight-line displacement from

(0,0)

(24, -7)

using the distance formula.

Distance

d = \sqrt{24^2 + (-7)^2} = \sqrt{576 + 49} = \sqrt{625} = 25\text{ km}

The shortest distance between origin and final point is given by

d = \sqrt{\Delta x^2 + \Delta y^2}

Key Concept

Direction and Distance Test involving multi-turn vector resolution, angular rotations, and Pythagorean straight-line distance calculation.

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