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The problem provided is as follows:

A man walks:
1. One mile east,
2. One mile northeast (at a 45-degree angle),
3. Then another mile east.

We need to find:
- The total distance between the man's initial and final positions.

Additionally:
- We need to compare this distance to another course where the man walks:
1. One mile east,
2. One mile north,
3. Then another mile east.

Let's solve this step by step.

### Path 1: East, Northeast, East
- **Step 1**: He walks 1 mile east, so his position is at (1, 0) in a coordinate plane.
- **Step 2**: Walking 1 mile northeast means moving at a 45° angle, so his displacement is \((\cos(45°), \sin(45°)) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), moving 1 mile means his displacement is \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\) in each direction.
  So, after this move, his new position is:
  \[
  \left(1 + \frac{\sqrt{2}}{2}, 0 + \frac{\sqrt{2}}{2}\right)
  = \left(1 + 0.707, 0.707\right)
  = (1.707, 0.707)
  \]
- **Step 3**: He walks another mile east, so his new position is \((1.707 + 1, 0.707) = (2.707, 0.707)\).

The final position after this path is approximately (2.707, 0.707). Now, we calculate the straight-line distance from the origin (0,0) to this point:
\[
d = \sqrt{(2.707)^2 + (0.707)^2} \approx \sqrt{7.329 + 0.5} = \sqrt{7.829} \approx 2.8 \text{ miles}.
\]

### Path 2: East, North, East
- **Step 1**: Walk 1 mile east to (1, 0).
- **Step 2**: Walk 1 mile north to (1, 1).
- **Step 3**: Walk another mile east, reaching (2, 1).

Now, calculate the distance from (0, 0) to (2, 1):
\[
d = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236 \text{ miles}.
\]

### Comparison:
- Distance for Path 1 (East, Northeast, East) = 2.8 miles.
- Distance for Path 2 (East, North, East) = 2.236 miles.

So, the first path (East, Northeast, East) is approximately **0.564 miles longer** than the second path (East, North, East).

Would you like further details or any clarifications?

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Here are 5 related questions for deeper understanding:
1. What is the geometric significance of walking northeast at 45 degrees?
2. How can you generalize this approach for different angles other than 45 degrees?
3. What would be the total distance if the person took 1 mile southwest instead of northeast in the first path?
4. How does the Pythagorean theorem apply in calculating these distances?
5. Could you compute the change in distances for other shapes of paths (e.g., triangular routes)?

**Tip**: In problems involving direction and distance, converting movements into coordinates simplifies calculations by allowing the use of the Pythagorean theorem.

A man walks one mile east, then one mile northeast, then another mile east. Find the distance, in miles, between the man’s initial and final positions. How much shorter or longer is the distance if the course is one mile east, one mile north, then one mile east?

Learn how to calculate the distance between initial and final positions of a man walking east, northeast, and then east using the distance formula and trigonometry. Compare it to an alternative path involving east and north directions. Suitable for students in Grades 9-11.

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