Mr. Yanki's Driving Problem: Finding the Return Path

This problem involves tracking movement based on directions and distances to find the shortest path back to the starting point. We can solve this by treating the movements as vectors or by using a coordinate system.

Understanding Mr. Yanki's Movements

Let's break down each step of Mr. Yanki's journey. We'll use a coordinate system where Point A is the origin (0, 0). We'll assume North corresponds to the positive y-axis and East corresponds to the positive x-axis. Therefore, South is the negative y-axis and West is the negative x-axis.

Tracking Position with Coordinates

We can track Mr. Yanki's position after each segment of his journey. A table helps visualize these changes.

Calculating Final Position

After all the movements, Mr. Yanki stops at Point P. Let's calculate the net displacement in the x (East-West) and y (North-South) directions:

• Net East-West Displacement: Sum of x-changes = $0 + (-19) + 0 + 15 + 0 + 6 + 0 = 2$ km. This means Point P is 2 km East of Point A.
• Net North-South Displacement: Sum of y-changes = $(-17) + 0 + 29 + 0 + (-1) + 0 + (-11) = 17 - 17 = 0$ km. This means Point P is on the same East-West line as Point A.

So, the final position of Point P relative to Point A is (2, 0).

Determining Return Path to Point A

Mr. Yanki is currently at Point P (2, 0) and needs to return to Point A (0, 0).

The required displacement vector from P to A is:

Displacement = (Target x - Current x, Target y - Current y)

Displacement = ($0 - 2$, $0 - 0$) = (-2, 0)

A coordinate change of (-2, 0) means moving 2 units in the negative x-direction.

• The negative x-direction corresponds to the West direction.
• The magnitude of the movement is $ \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2 $ km.

Therefore, Mr. Yanki needs to drive 2 km towards the West to reach Point A again.