Problem Breakdown:

We are tasked with finding the difference in distance between two routes:

1. Jeb's Route:

   • Jeb walks along Sono Road from $P$ to $X$, a distance of 30 meters.
   • Then, he turns east and walks 20 meters along Ewest Road from $X$ to $Q$.
   • Total distance = $30 + 20 = 50$ meters.

2. Jill's Route:

   • Jill walks on the straight diagonal path from $P$ to $Q$.

We need to calculate Jill's distance and subtract it from Jeb's total distance to find the difference.

Solution:

Using the Pythagorean Theorem, the straight-line distance Jill travels ($PQ$) is calculated as follows:

$PQ = \sqrt{(PX)^2 + (XQ)^2}$

Where:

• $PX = 30$ meters (vertical distance along Sono Road).
• $XQ = 20$ meters (horizontal distance along Ewest Road).

Substitute the values:

$PQ = \sqrt{30^2 + 20^2}$

$PQ = \sqrt{900 + 400} = \sqrt{1300} = 10\sqrt{13} \approx 36.06 \, \text{meters}$

Difference in Distance:

Jeb's distance = $50 \, \text{meters}$
Jill's distance = $36.06 \, \text{meters}$

Difference: $50 - 36.06 = 13.94 \, \text{meters}$

Final Answer:

The difference in distance between the two routes is approximately:

$\boxed{13.94 \, \text{meters}}$

Would you like me to explain further or explore related topics? Here are some related questions you might find useful:

1. How does the Pythagorean theorem work in other coordinate systems?
2. What is the significance of straight-line distances in optimization problems?
3. Could Jill's path be calculated using other mathematical techniques, like vectors?
4. How is the Pythagorean theorem used in real-world applications, like navigation or construction?
5. What would the difference be if Jeb walked a different route, e.g., along a curved path?

Tip: For similar problems involving distances, always visualize the situation geometrically, and check if straight-line distances can be solved with the Pythagorean theorem!