Problem 57 - Algebra and Trigonometry

Question

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 4th St. and 2nd Ave. and the corner of 11th St. and 26th 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) Points must be in a straight line.

Step 1: Understand the Problem

To solve the given problem, we are looking to find the straight-line distance (using the Euclidean metric) between two points on a grid, given their walking distances. We also need to determine specific conditions for when the walking distance equals the straight-line distance.

Step 2: Walking Distance to Straight-Line Distance

To find the straight-line distance, we use the Pythagorean theorem when the walking distance (grid-like movement) is given. The formula to compute the straight-line distance \(d\) is given by: \[ d = \sqrt{a^2 + b^2} \]where \(a\) and \(b\) are the number of blocks in the east-West and north-south directions respectively.

Step 3: Calculate the Straight-Line Distance for Part (a)

Given that the walking distance from A to B is 3 blocks east and 4 blocks north, we use the formula to find the straight-line distance:\[ d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] blocks.Therefore, the straight-line distance between A and B is 5 blocks.

Step 4: Calculate Distances for Part (b)

For the streets, 4th St. to 11th St. is 7 blocks (north-south) and from 2nd Ave. to 26th Ave. is 24 blocks (east-west). 1. **Walking Distance:** 7 blocks + 24 blocks = 31 blocks.2. **Straight-Line Distance:** \[ d = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \] blocks.Hence, the walking distance is 31 blocks, and the straight-line distance is 25 blocks.

Step 5: Condition for Equality Between Walking and Straight-Line Distances

The walking distance equals the straight-line distance if the path does not need to make any turns—meaning, the path is a straight line. This is true when points P and Q are either directly horizontal or directly vertical to each other, traversing either solely along a single street or avenue without change in direction.

Key Concepts

Euclidean metric

The Euclidean metric is a way to measure straight-line distances between two points in a plane. It is derived from a familiar concept in everyday life, such as measuring the shortest possible path between two locations. This method requires us to consider only the direct distance, not what routes or obstacles might exist in between. By using the Euclidean metric, we are focusing on the "as-the-crow-flies" distance, which simplifies our calculations by removing any detours or turns. This is especially useful in mathematical contexts where we are interested in purely theoretical distances. When applying the Euclidean metric in a grid-like layout such as a city grid, it allows us to find this shortest path using formulaic methods."
The primary tool to apply the Euclidean metric in practice is the Distance Formula, which we will explore next, as it aligns directly with this concept."

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, typically used to determine the length of the sides of right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.This theorem is crucial for calculating straight-line distances on a grid using the Euclidean metric. In practical terms, if you know the number of blocks (or units) moved horizontally and vertically in a grid, you can determine the straight-line distance by treating these as the two shorter sides of a right triangle.

• Given sides: 3 blocks east and 4 blocks north.
• Apply the theorem: \(c = \sqrt{a^2 + b^2}\).
• Calculation: \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) blocks.

This method is both simple and profound, highlighting the geometric beauty of spatial relationships through simple numeric solutions."
The next step is to understand how this applies specifically in city grids, often leading to the concept of straight-line distances."

straight-line distance

In a city with a grid-like pattern of streets and avenues, the straight-line distance refers to the shortest possible path between two points measured directly across space. Unlike walking distance, straight-line distance does not account for the path one would take through streets and alleys.Finding the straight-line distance is key when one wishes to quickly determine how far apart two points are without any detours. Calculating this distance can easily be done using the Distance Formula derived from the Pythagorean theorem: \(d = \sqrt{a^2 + b^2} \), where \(a\) and \(b\) are the horizontal and vertical components (or the changes in x and y). For example, in a scenario where you need to find the distance between two points, say 4th St. and 2nd Ave. to 11th St. and 26th Ave., the calculation is as follows:

• 7 blocks north-south, 24 blocks east-west.
• Straight-Line Distance: \(\sqrt{7^2 + 24^2} = 25\) blocks.

Straight-line distance helps in understanding the geometric layout of paths and is often more intuitive in certain problem-solving scenarios."
We now consider the notion of walking distance, which contrasts with this concept."

walking distance

Walking distance refers to the actual number of blocks one would have to travel on a grid-like pattern of streets and avenues to move from one point to another. Unlike straight-line distance, it accounts for the physical constraints of movement, such as having to follow the layout of streets and not being able to cut through buildings or obstacles directly. This measure is realistic for real-world navigation and urban planning, where straight paths are rarely possible through obstructions. Walking distance is typically greater than straight-line distance for this reason, as you need to make turns at intersections and follow the grid layout. In scenarios like moving from 4th St. and 2nd Ave. to 11th St. and 26th Ave., the walking distance would be:

• 7 blocks north, then 24 blocks east = 31 blocks in total walking distance.

Understanding walking distance is essential when considering the practical path followed in navigational contexts, making it distinctively different from the more theoretical concept of straight-line distance. The relationship between these two distances offers insight into navigational efficiencies or costs in real urban settings."
Each distance measure serves its purpose, whether conceptual or practical, and highlights how movement is influenced by both geometric and real-world factors."