A geological survey team is navigating an uncharted terrain using a miscalibrated digital … — State PSC Exam

A geological survey team is navigating an uncharted terrain using a miscalibrated digital compass. The compass incorrectly displays 'North' when it is actually pointing towards true North-West.

According to the readings on this faulty compass, the team leader executes the following sequence of movements:
- First, he travels $12\sqrt{2}$ km towards the 'East' indicated by the compass.
- Second, he rotates $135^\circ$ to his right and travels $12$ km straight.
- Finally, he rotates $90^\circ$ to his left and travels another $12$ km.

What is the team's shortest straight-line distance and true geographic direction from their starting point?

A
$24$ km, True South-East
B
$24$ km, True South
$24$ km, True EastAnswer
D
$24$ km, True North

Answer

The team's shortest straight-line distance is

24

km, and their true geographic direction from the starting point is True East.

The miscalibrated compass means all readings are shifted. Compass North is True North-West, so Compass East is True North-East. A

12\sqrt{2}

km movement True North-East equals

(12, 12)

on a coordinate plane. A

135^\circ

right turn from Compass East faces Compass South-West, which is True South. Moving

12

km True South changes the position to

(12, 0)

. A

90^\circ

left turn from Compass South-West faces Compass South-East, which is True East. Moving

12

km True East changes the position to

(24, 0)

. This represents a distance of

24

km in the True East direction.

Step-by-Step Solution

Determine the true geographic bearing for the initial movement.

The compass reads 'North' when pointing True North-West. This means true directions are

45^\circ

counter-clockwise from compass readings. The team moves towards Compass 'East', which corresponds to True North-East. The displacement of

12\sqrt{2}

km True North-East translates to

12

km True East and

12

km True North.

Establishing the relationship between the faulty compass and true geographic directions is required to map the exact path correctly.

Calculate the true bearing and displacement for the second movement.

Turning

135^\circ

right (clockwise) from Compass East results in facing Compass South-West. Compass South-West corresponds to True South. Moving

12

km True South cancels out the previous

12

km True North, placing the team at exactly

12

km True East from the starting point.

Angular turns apply to the current facing direction. A

135^\circ

clockwise rotation from East lands on South-West.

Calculate the true bearing and displacement for the final movement.

Turning

90^\circ

left (counter-clockwise) from Compass South-West results in facing Compass South-East. Compass South-East corresponds to True East. Moving

12

km True East adds to the previous position, resulting in a total displacement of

24

km True East.

90^\circ

counter-clockwise rotation from South-West lands on South-East. Summing the true vectors provides the final destination.

Key Concept

Vector Addition and Relative Bearings in Miscalibrated Systems

Question

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Step-by-Step Solution

Key Concept

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