Question

The figure shows an urban road map of a city. The boundary of the city is a perfect rectangle as indicated by the black dotted line. The grey lines indicate the major roads that run parallel to the edges of the city. The red line shows the route taken by a bus from point P to point Q. If the perimeter of the boundary is 68 km, what is the distance travelled by the bus in kilometres? 

Hint:

For any path on a grid that moves from one corner of a rectangle to the opposite corner without backtracking (i.e., moving only right/left and up/down in a consistent direction), the total path length is always the same: Length + Width of the rectangle. This is half the perimeter.

Answer 1

Step 1: Understanding the Concept:
The problem asks for the length of a specific path (the bus route) on a grid. The path is constrained to move along grid lines within a rectangle of a given perimeter. The key insight is to relate the total path length to the dimensions of the rectangle.
Step 2: Key Formula or Approach:
Let the length of the rectangular boundary be \(L\) and the width be \(W\).
The perimeter of a rectangle is given by the formula: \(P = 2(L + W)\).
The bus route from P to Q is a "rectilinear" or "Manhattan" path. The total distance of such a path is the sum of its total horizontal movement and its total vertical movement.
Step 3: Detailed Explanation:


Use the given perimeter: We are given that the perimeter of the boundary is 68 km. \[ 2(L + W) = 68 \text{ km} \] Dividing by 2, we get the sum of the length and width: \[ L + W = 34 \text{ km} \]
Analyze the bus route:

Point P is at one corner of the rectangle (e.g., the bottom-left).
Point Q is at the opposite corner (the top-right).
To travel from P to Q by only moving along the grid lines, the bus must cover a total horizontal distance equal to the length of the rectangle, \(L\).
Similarly, the bus must cover a total vertical distance equal to the width of the rectangle, \(W\).
The red line shows a path made up of many horizontal and vertical segments. The sum of the lengths of all horizontal segments equals \(L\), and the sum of the lengths of all vertical segments equals \(W\), because the path is always progressing towards Q without backtracking.

Calculate the total distance: The total distance travelled by the bus is the sum of the total horizontal and vertical distances. \[ \text{Total Distance} = (\text{Total Horizontal Travel}) + (\text{Total Vertical Travel}) \] \[ \text{Total Distance} = L + W \]
Substitute the value from Step 1: From our perimeter calculation, we know that \(L + W = 34\) km. Therefore, the distance travelled by the bus is 34 km.

Step 4: Final Answer:
The distance travelled by the bus is 34 kilometres.

Answer 2

Let the rectangle's length be \( L \) and width be \( W \), so \( 2(L+W) = 68 \), which gives \( L + W = 34 \) km. To see why the bus travels exactly this distance, it helps to walk through one concrete example rather than argue in general terms.

Suppose, for instance, that the bus's actual route from P to Q happens to go right 6 km, up 4 km, right 9 km, up 5 km, right 5 km, then up 5 km, before reaching Q. Adding the rightward pieces: \( 6 + 9 + 5 = 20 \) km, and this must equal the rectangle's length \( L \). Adding the upward pieces: \( 4 + 5 + 5 = 14 \) km, and this must equal the width \( W \).

Check: \( L + W = 20 + 14 = 34 \) km, which matches what the perimeter gives us, no matter how the individual zig-zag segments are drawn, since every rightward piece is part of the same total length \( L \), and every upward piece is part of the same total width \( W \).

So the correct answer is 34 kilometres.