Determining Final Distance from Starting Point

This problem involves calculating the net displacement of a person after a series of movements in different directions. We need to find the straight-line distance between the final position and the initial starting point.

Analyzing the Movements

Let's break down the movements step-by-step:

• The person starts at a point (let's call it the origin).
• Movement 1: Travels 12 km North.
• Movement 2: Turns East and walks 5 km.
• Movement 3: Turns South and walks 3 km.
• Movement 4: Turns West and walks 5 km.

Calculating Net Displacement

We can calculate the total displacement in the North-South direction and the East-West direction separately.

North-South Displacement:

• The person moves 12 km North (positive direction).
• Later, the person moves 3 km South (negative direction).
• Net North-South displacement = 12 km (North) - 3 km (South)
• Using LaTeX for clarity: \(12 \text{ km (North)} - 3 \text{ km (South)} = 9 \text{ km North}\)

East-West Displacement:

• The person moves 5 km East (positive direction).
• Later, the person moves 5 km West (negative direction).
• Net East-West displacement = 5 km (East) - 5 km (West)
• Using LaTeX: \(5 \text{ km (East)} - 5 \text{ km (West)} = 0 \text{ km}\)

Final Position and Distance

The net displacement is 9 km North and 0 km East-West. This means the person's final position is exactly 9 km directly North of the starting point.

The distance from the starting point is the magnitude of this net displacement. Since the East-West component is zero, the distance is simply the North-South component.

Distance = \(\sqrt{(\text{Net East-West Displacement})^2 + (\text{Net North-South Displacement})^2}\)

Distance = \(\sqrt{(0 \text{ km})^2 + (9 \text{ km})^2}\)

Distance = \(\sqrt{0 + 81 \text{ km}^2}\)

Distance = \(\sqrt{81 \text{ km}^2}\)

Distance = \(9 \text{ km}\)

Therefore, the person is 9 km from the starting point.