Flag question: Question 13

Question 131 **pts**

Morgan’s office, shown in the diagram, is **10 miles** from his home. The question marks represent possible locations of a pizza parlor, which is *x* miles from Morgan’s home.

On Fridays Morgan treats himself to pizza for lunch, so over the course of the day he drives from his home to work, from work to the pizza parlor, back to work, then from work back to his home at the end of the day. The day’s trip is a total of **30 miles**.

Indicate the **shortest possible distance** from the pizza parlor to Morgan's home.

To solve this problem, let's break it down step by step:

1. **Total Distance Travelled:**
   - Morgan’s total distance for the day is **30 miles**. This includes:
     - Driving from **home to work** (10 miles),
     - Driving from **work to the pizza parlor** and back to work (a round trip),
     - Finally, driving from **work to home** (10 miles).

2. **Distance Breakdown:**
   - The distance from home to work and from work back home is fixed at **10 miles each**, making the total **20 miles** just for the commute to and from work.
   - This leaves **30 - 20 = 10 miles** for the round trip from work to the pizza parlor and back to work.

3. **Distance to Pizza Parlor:**
   - Since the distance from work to the pizza parlor and back to work totals **10 miles**, the one-way distance from work to the pizza parlor is half of that, which is **5 miles**.

4. **Shortest Distance from Home to Pizza Parlor:**
   - The pizza parlor forms a triangle with Morgan’s home and work. Morgan's home is **10 miles** from his work, and the pizza parlor is **5 miles** from his work. 
   - The shortest possible distance from Morgan's home to the pizza parlor can be found using the **triangle inequality** or by checking if it's possible to use the pizza parlor’s proximity to work.

Let’s now use the Pythagorean theorem to find the shortest distance:
- One leg of the triangle is **10 miles** (home to work).
- The other leg is **5 miles** (work to pizza parlor).
  
The shortest distance \(d\) from the home to the pizza parlor is the hypotenuse of this right triangle:
\[
d = \sqrt{(10)^2 + (5)^2} = \sqrt{100 + 25} = \sqrt{125} \approx 11.18 \text{ miles}
\]

### Conclusion:
The **shortest possible distance** from Morgan's home to the pizza parlor is approximately **11.18 miles**.

Would you like further details or have any questions?

Here are some related questions:
1. How would the solution change if the pizza parlor were closer to Morgan’s work?
2. What if the total distance for the day was 35 miles instead of 30 miles?
3. Could there be another configuration where the pizza parlor is farther away from work but still satisfies the total trip?
4. How would you calculate the shortest distance if Morgan took a different route to the pizza parlor?
5. How does this problem change if the distance between home and work were different?

**Tip:** In any triangle, the triangle inequality can help estimate possible side lengths based on given distances!

Morgan’s office is 10 miles from his home. The question marks represent possible locations of a pizza parlor, which is x miles from Morgan’s home. On Fridays, Morgan treats himself to pizza for lunch, so over the course of the day he drives from his home to work, from work to the pizza parlor, back to work, and then from work back to his home at the end of the day. The day’s trip is a total of 30 miles. Indicate the shortest possible distance from the pizza parlor to Morgan's home.

Find the shortest possible distance from Morgan's home to the pizza parlor using the Pythagorean theorem. This geometry problem involves calculating distances, understanding triangle inequality, and applying the Pythagorean theorem. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation