Let's analyze the problem step by step.
Problem Restatement
- A boy rides his bicycle northward initially for some distance xxx km.
- Then he turns left and rides 1 km.
- Then he turns left again and rides 2 km.
- After these moves, he is 1 km west of his starting point.
- We need to find how far he rode northward initially (i.e., find xxx).
Step-by-Step Solution
Step 1: Define directions and movements
- The boy starts at point OOO.
- He rides northward xxx km. So his position is now (0,x)(0,x)(0,x).
- He turns left. Since he was going north, turning left means going west.
- He rides 1 km west. Position now: (−1,x)(-1,x)(−1,x).
- He turns left again. From west, turning left means going south.
- He rides 2 km south. Position now: (−1,x−2)(-1,x-2)(−1,x−2).
Step 2: Use the final position information
The problem states he is 1 km west of his starting point after these moves. The starting point is at (0,0)(0,0)(0,0), so the final position is (−1,y)(-1,y)(−1,y) for some yyy. From Step 1, the final position is (−1,x−2)(-1,x-2)(−1,x−2). Since he is 1 km west of the starting point, the x-coordinate is indeed −1-1−1, which matches.
Step 3: Determine the north-south displacement
The problem doesn't explicitly say how far north or south he is relative to the starting point, only that he is 1 km west. However, since the problem is about how far he rode north initially, and the final vertical position is x−2x-2x−2, the vertical displacement from the starting point is x−2x-2x−2. The problem does not say he ended up at the same latitude as the start, so the vertical displacement can be anything.
Step 4: Check for any additional constraints
The problem states: "He found himself 1 km west of his starting point." This means the final position is exactly 1 km west, but no mention of north or south displacement. Therefore, the final position is (−1,y)(-1,y)(−1,y), where y=x−2y=x-2y=x−2. Since the problem is consistent with this, the only unknown is xxx.
Step 5: Consider the possibility that the final position is 1 km west and
0 km north/south (i.e., same latitude)
If the problem implies he is exactly 1 km west of the starting point (same latitude), then:
x−2=0 ⟹ x=2x-2=0\implies x=2x−2=0⟹x=2
Final answer:
The boy initially rode 2 km northward.
Summary:
Movement| Position (x, y)
---|---
Start| (0, 0)
After riding north xxx km| (0, xxx)
After turning left (west) and riding 1 km| (-1, xxx)
After turning left (south) and riding 2 km| (-1, x−2x-2x−2)
Given final position is 1 km west of start (i.e., −1,0-1,0−1,0):
x−2=0 ⟹ x=2x-2=0\implies x=2x−2=0⟹x=2
If you have any further questions or want a visual diagram, feel free to ask!
