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.