https://file.aiquickdraw.com/mathbot/file/172957020670492y5tad5.jpg

On the map​ (right), the length of each​ east-west block is one fifth
 
mile and the length of each​ north-south block is one fourth
 
mile. Victoria has to walk from the bus stop
 
to the theater
. 
Find the shortest walking distance. Then find the​ straight-line distance​ ('as the crow​ flies') between the two locations.
Question content area bottom left
Part 1
What is the shortest walking​ distance?
enter your response here
 
mi
​(Round to the nearest hundredth as​ needed.)


Let's break down the problem:

### Step 1: Understanding the grid distances
- Each **east-west block** is given as **\(\frac{1}{5}\) mile**.
- Each **north-south block** is given as **\(\frac{1}{4}\) mile**.

We need to find:
1. The **shortest walking distance** (which means moving along the grid).
2. The **straight-line distance** (which is a direct diagonal from the bus stop to the theater).

### Step 2: Determining the grid locations
From the map:
- The **bus stop** is at the bottom left.
- The **theater** is at the top left.

Counting the blocks:
- **Horizontal distance** (east-west): There seems to be **2 blocks** between the bus stop and the theater.
- **Vertical distance** (north-south): There are **4 blocks** between the bus stop and the theater.

### Step 3: Finding the shortest walking distance
To find the shortest walking distance, we will sum the distance traveled along the east-west and north-south directions:
\[
\text{Walking Distance} = \text{Horizontal distance} + \text{Vertical distance}
\]
- Horizontal distance: \(2 \text{ blocks} \times \frac{1}{5} \text{ mile/block} = \frac{2}{5} = 0.4 \text{ miles}\)
- Vertical distance: \(4 \text{ blocks} \times \frac{1}{4} \text{ mile/block} = 1 \text{ mile}\)

Thus, the total walking distance is:
\[
\text{Walking Distance} = 0.4 \, \text{miles} + 1 \, \text{mile} = 1.4 \, \text{miles}
\]

### Step 4: Finding the straight-line distance (as the crow flies)
The straight-line distance forms a right triangle, where:
- One leg is the horizontal distance: \(0.4\) miles.
- The other leg is the vertical distance: \(1\) mile.

Using the Pythagorean Theorem to find the hypotenuse (straight-line distance):
\[
\text{Straight-line Distance} = \sqrt{(0.4)^2 + (1)^2}
\]
\[
\text{Straight-line Distance} = \sqrt{0.16 + 1} = \sqrt{1.16} \approx 1.08 \, \text{miles}
\]

### Final answers:
1. The **shortest walking distance** is **1.4 miles**.
2. The **straight-line distance** (as the crow flies) is **1.08 miles**.

Would you like further details or clarifications on any part?

Here are 5 related questions:
1. How does using the Pythagorean Theorem help in finding straight-line distances in grids?
2. What if the theater were moved 1 block east? How would the distances change?
3. Can you think of real-life applications where grid distances vs. straight-line distances are important?
4. What is the formula for distance in a grid-like city (Manhattan distance)?
5. Why is the straight-line distance always shorter than the grid distance?

**Tip:** The shortest walking distance in a grid is often referred to as the "Manhattan distance," as it follows the grid-like pattern of streets in cities like Manhattan, NY.

On the map​ (right), the length of each​ east-west block is one fifth mile and the length of each​ north-south block is one fourth mile. Victoria has to walk from the bus stop to the theater. Find the shortest walking distance. Then find the​ straight-line distance​ ('as the crow​ flies') between the two locations.

Learn how to find the shortest walking distance and straight-line distance between two points on a city grid map. This example covers the use of Manhattan distance for walking routes and the Pythagorean theorem for diagonal distances, with measurements in miles. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation