A field researcher starts from a campsite and walks 10\text{ km} due North. She then turns… — State PSC Exam

A field researcher starts from a campsite and walks $10\text{ km}$ due North. She then turns right and walks $7\text{ km}$ due East. Next, she turns left and walks $6\text{ km}$ due North. Finally, she turns right and walks $5\text{ km}$ due East. What is the shortest straight-line distance (in km) from her final position to the campsite?

Answer: 20 km

Answer

The shortest straight-line distance from the final position to the campsite is 20 km.

The researcher ends up

16\text{ km}

North and

12\text{ km}

East of her starting position. Using the right-triangle distance formula

\sqrt{\text{North}^2 + \text{East}^2}

, we get

\sqrt{16^2 + 12^2} = \sqrt{400} = 20\text{ km}

Step-by-Step Solution

Sum up the individual directional movements along the North-South axis.

Total North displacement =

10\text{ km} + 6\text{ km} = 16\text{ km North}

Both vertical moves are towards the North, so their distances add together directly.

Sum up the individual directional movements along the East-West axis.

Total East displacement =

7\text{ km} + 5\text{ km} = 12\text{ km East}

Both horizontal moves are towards the East, so their distances add together directly.

Calculate the hypotenuse using the Pythagorean theorem.

Shortest distance =

\sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20\text{ km}

The net North and East displacements form a right-angled triangle where the shortest distance is the hypotenuse.

Key Concept

Displacement Vector Summation and Pythagorean Distance Calculation

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