https://file.aiquickdraw.com/mathbot/file/1727839596535cb5cacy0.jpg

We are tasked with determining a route that covers all the locations—Post Office, Bank, and Dry Cleaners—while keeping the total distance traveled under 14.5 miles. 

The triangle on the diagram provides the following distances:
- Home to Bank: 2.5 miles
- Bank to Dry Cleaners: 3 miles
- Dry Cleaners to Home: 5 miles
- Post Office to Bank: 2 miles
- Post Office to Dry Cleaners: 4 miles
- Home to Post Office: 5 miles

Let’s now compute the total distance for each route provided:

**Option A: Home → Post Office → Dry Cleaners → Bank → Home**
- Home to Post Office: 5 miles
- Post Office to Dry Cleaners: 4 miles
- Dry Cleaners to Bank: 3 miles
- Bank to Home: 2.5 miles
- Total = 5 + 4 + 3 + 2.5 = 14.5 miles

**Option B: Home → Bank → Dry Cleaners → Post Office → Home**
- Home to Bank: 2.5 miles
- Bank to Dry Cleaners: 3 miles
- Dry Cleaners to Post Office: 4 miles
- Post Office to Home: 5 miles
- Total = 2.5 + 3 + 4 + 5 = 14.5 miles

**Option C: Home → Post Office → Bank → Dry Cleaners → Home**
- Home to Post Office: 5 miles
- Post Office to Bank: 2 miles
- Bank to Dry Cleaners: 3 miles
- Dry Cleaners to Home: 5 miles
- Total = 5 + 2 + 3 + 5 = 15 miles

**Option D: Home → Bank → Post Office → Dry Cleaners → Home**
- Home to Bank: 2.5 miles
- Bank to Post Office: 2 miles
- Post Office to Dry Cleaners: 4 miles
- Dry Cleaners to Home: 5 miles
- Total = 2.5 + 2 + 4 + 5 = 13.5 miles

**Option E: Home → Dry Cleaners → Post Office → Bank → Home**
- Home to Dry Cleaners: 5 miles
- Dry Cleaners to Post Office: 4 miles
- Post Office to Bank: 2 miles
- Bank to Home: 2.5 miles
- Total = 5 + 4 + 2 + 2.5 = 13.5 miles

**Conclusion:**
- The two routes that have a total distance less than 14.5 miles are **Option D** and **Option E**. 

Both options result in a total of 13.5 miles. Therefore, **D and E** are valid answers.

Would you like further clarification or more details?

Here are five related questions for further exploration:
1. How can we use geometry to calculate the shortest route between multiple points?
2. What are other methods to minimize travel distances in a network?
3. How can the triangle inequality theorem help in determining the shortest path in a geometric figure?
4. How can a route optimization algorithm like Dijkstra’s or the Traveling Salesman Problem be applied here?
5. How would the problem change if the distances between the points were unknown or irregular?

**Tip:** In route optimization, understanding the relationship between points and distances can help in minimizing time and cost, which is why tools like mapping algorithms are useful!

Determine a route whose distance is less than 14.5 miles for running errands to all three destinations and returning home. Choose the route that is less than 14.5 miles.

Learn how to determine the shortest route covering multiple destinations while keeping the total distance less than 14.5 miles. This problem involves calculating distances and using the triangle inequality theorem. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation