Analyzing Tanisha's Driving Route

The problem requires calculating the net displacement to find the shortest distance and direction from Tanisha's final stop (Point P) back to the starting point (Point A).

Mapping the Journey

We can track Tanisha's position relative to Point A using coordinates, where East is the positive x-axis and North is the positive y-axis. Point A is at the origin (0, 0).

• Starts at A: (0, 0)
• Drives 68 km East: Position is (68, 0).
• Turns left (North), drives 45 km: Position becomes (68, 0 + 45) = (68, 45).
• Turns left (West), drives 78 km: Position becomes (68 - 78, 45) = (-10, 45).
• Turns left (South), drives 71 km: Position becomes (-10, 45 - 71) = (-10, -26).
• Turns left (East), drives 10 km: Position becomes (-10 + 10, -26) = (0, -26).
• Stops at Point P: (0, -26).

Calculating Net Displacement

The net displacement calculation simplifies the movements:

• Total East-West movement: \(68 \text{ km (East)} - 78 \text{ km (West)} + 10 \text{ km (East)} = (68 - 78 + 10) \text{ km} = 0 \text{ km}\).
• Total North-South movement: \(45 \text{ km (North)} - 71 \text{ km (South)} = (45 - 71) \text{ km} = -26 \text{ km}\).

A net movement of 0 km in the East-West direction means Point P is directly North or South of the initial East-West line through A. The net movement of -26 km in the North-South direction means Point P is 26 km South of A's latitude.

Therefore, Point P is located at (0, -26) relative to A at (0, 0).

Finding the Return Path from P to A

To travel from Point P (0, -26) back to Point A (0, 0), Tanisha needs to cover the following change in coordinates:

• Change in X-coordinate: \(0 - 0 = 0\) km.
• Change in Y-coordinate: \(0 - (-26) = +26\) km.

This means Tanisha must travel 26 km in the positive Y direction, which is North.

The shortest distance is 26 km and the direction is North.