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Difficulty: MediumDirection and Distance Test

A maritime patrol boat departs from its home harbor to secure coastal waters. It first sails 24 km24\text{ km} strictly towards the West to investigate a radar blip. Finding nothing, it turns North and travels 7 km7\text{ km} to a reported location. It then changes course, sailing 45 km45\text{ km} East to follow a distress signal. Finally, it turns South and travels 27 km27\text{ km} to successfully intercept the target. Calculate the exact straight-line distance from the home harbor to the interception point.

Answer: 29 km

Answer

29
By resolving the boat's path into net vertical and horizontal components, we find its final position is 21 km21\text{ km} East and 20 km20\text{ km} South of its starting point. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), the shortest straight-line distance is the hypotenuse: 212+202=29 km\sqrt{21^2 + 20^2} = 29\text{ km}.

Step-by-Step Solution

1
Calculate the net horizontal (East-West) displacement.
21 km21\text{ km} East
The boat initially travels 24 km24\text{ km} West, then later travels 45 km45\text{ km} East. The net horizontal movement is 45−24=21 km45 - 24 = 21\text{ km} in the East direction.
2
Calculate the net vertical (North-South) displacement.
20 km20\text{ km} South
The boat travels 7 km7\text{ km} North, and later travels 27 km27\text{ km} South. The net vertical movement is 27−7=20 km27 - 7 = 20\text{ km} in the South direction.
3
Apply the Pythagorean theorem to find the shortest straight-line distance.
212+202=441+400=841=29 km\sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29\text{ km}
The net East and South displacements form the two perpendicular legs of a right-angled triangle relative to the starting point. The straight-line distance is the hypotenuse.

Key Concept

Vector displacement across cardinal directions and Pythagorean theorem calculation.
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Topics

Direction and Distance Test

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