Problem 51 - Algebra and Trigonometry

Question

Distances in a City \(\quad\) A city has streets that run north and south, and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points \(A\) and \(B\) is 7 blocks \(-\) that is, 3 blocks east and 4 blocks north. To find the straight-line distances \(d,\) we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between \(A\) and \(B\) . (b) Find the walking distance and the straight-line distance between the corner of 4 th St. and 2 nd Ave. and the corner of 11 th St. and 26 th Ave. (c) What must be true about the points \(P\) and \(Q\) if the walking distance between \(P\) and \(Q\) equals the straight-line distance between \(P\) and \(Q ?\)

Step-by-Step Solution

Answer

(a) 5 blocks. (b) Walking: 31 blocks; Straight-line: 25 blocks. (c) P and Q must be on the same line with no turns.

Step 1: Identify Coordinates of Points

First, we identify the coordinates for points A and B. Given the problem, assume point A is at (0,0) and point B is at (3,4), where the numbers represent blocks. In part (b), identify coordinates as (2,4) for 4th St. and 2nd Ave., and (26,11) for 26th Ave. and 11th St.

Step 2: Apply the Distance Formula

The Distance Formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Calculate for each pair of coordinates. For part (a), the straight-line distance is: \[ d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] blocks.

Step 3: Calculate Walking Distance

Walking distance is the sum of block movements east/west and north/south. For (2,4) to (26,11): - Move horizontally from 2nd Ave. to 26th Ave.: 24 blocks. - Move vertically from 4th St. to 11th St.: 7 blocks. Thus, the walking distance is 24 + 7 = 31 blocks.

Step 4: Apply Distance Formula (Part b)

Using the coordinates (2,4) and (26,11) in the Distance Formula:\[ d = \sqrt{(26-2)^2 + (11-4)^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \] blocks.

Step 5: Understanding Equal Distances

In part (c), the walking and straight-line distances are equal when movement along the streets forms a diagonal, i.e., points P and Q are on the same linear diagonal in a grid. This occurs when either only east-west blocks or only north-south blocks are traveled, such as moving directly along a single street or avenue without turning.

Key Concepts

Straight-Line Distance

When referring to straight-line distance, it represents the shortest possible path between two points. Imagine drawing a direct line from one location to another on a map.
In coordinate geometry, the straight-line distance between two points can be calculated using the Distance Formula.
This formula, \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, \] makes use of the coordinates of each point to find the exact straight-line measurement.

• It calculates the direct distance across diagonal lines in a grid.
• Provides an efficient route if physical barriers are absent.
• Serves as a base measurement in many applications such as physics and navigation.

Straight-line distance is often less than the walking distance, as it does not require adherence to physical street layouts.

Walking Distance

Walking distance considers the actual traveled path following streets and avenues.
In a typical grid-like city layout, this involves moving in blocked segments, not diagonally. Unlike the straight-line distance, walking distance must align with existing pathways.

• Walking distance = Sum of blocks moved east/west + Sum of blocks moved north/south.
• This concept is crucial for understanding urban geography.
• It represents real-world conditions people face in urban settings.

For example, to get from a point on 4th St. to a corner on 26th Ave, you must account for how streets and avenues are structured. Therefore, one needs to calculate the total number of blocks traveled in each direction.

Coordinate Geometry

Coordinate geometry is a key mathematical tool used to study shapes, lines, and spaces via a numerical mapping system. It employs the concept of coordinates to represent different points in a space.
These coordinates are usually written as \((x, y)\), making it possible to calculate distances and slopes efficiently.

• It provides a systematic way to define locations geometrically.
• Helps in solving geometric problems by translating them into algebraic ones.
• Enables the graphical representation of equations and inequalities.

In a city grid, coordinate geometry helps logically layout intersections of streets and avenues, paving the way for straightforward problem-solving strategies using math. It simplifies finding relations between geometric problems and their algebraic solutions.

Distance Calculation

Distance calculation is the process of determining the measurable length between two points.
This can be done in various ways, depending on the specifics such as path and environment considered. The primary objectives in distance calculation involve:

• Choosing the right method: straight-line vs. walking distance.
• Understanding context: real-world vs. ideal conditions.
• Applying formulas correctly to obtain accurate results.

In the example of the city layout given, walking distance considers grid constraints by moving along streets and avenues, whereas straight-line distance demonstrates the unaltered direct span. By analyzing both measurements, one gains an appreciation of practical versus theoretical aspects of travel.