This problem is about finding the shortest possible swim path between the ends of two piers while making one stop on the beach, which leads to a classic "reflection" problem in geometry, often solved using principles similar to the Fermat point or the law of reflection.

Setting up the scenario:

1. You have two piers extending perpendicular into the ocean.
   • The longer pier extends 2 miles into the ocean.
   • The shorter pier extends 1 mile into the ocean.
2. The distance along the beach between the two piers is 4 miles.
3. The swimmer needs to swim from the end of the longer pier to the end of the shorter pier with one rest stop on the beach.

To minimize the swim distance, the key is to "reflect" the end of the shorter pier across the beach, turning the problem into a straight-line path.

Solution Steps:

1. Reflect the shorter pier: Imagine reflecting the end of the shorter pier across the beach (i.e., placing a point directly across the shore, 1 mile on the opposite side of the shore).

2. Straight-line distance: Now, instead of thinking about the swimmer stopping on the beach, imagine they swim directly to the reflected shorter pier. By doing so, we can treat the total path as a straight line from the end of the longer pier to the reflected end of the shorter pier. The point where this straight line crosses the beach will be the optimal rest stop.

3. Coordinates:

   • Assume the origin (0, 0) is the point where the longer pier meets the shore.
   • The end of the longer pier is at (0, 2).
   • The shorter pier meets the shore at (4, 0), and its end is at (4, 1).
   • The reflection of the shorter pier's end would be at (4, -1).

4. Minimize the swim path: To find the shortest path, calculate the straight-line distance from the end of the longer pier (at (0, 2)) to the reflected point (4, -1).

   The total distance is the length of the line segment from $(0, 2)$ to $(4, -1)$, given by the distance formula: $d = \sqrt{(4 - 0)^2 + (-1 - 2)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ miles}.$

Conclusion:

The shortest possible swim distance is 5 miles, including both swimming legs (from the longer pier to the beach and from the beach to the shorter pier).

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Would you like a more detailed breakdown of the reflection method or any additional questions about this solution?

Here are five related questions you could explore:

1. How would the solution change if the shore distance between the piers was different?
2. What if both piers were of the same length? How would the optimal path be affected?
3. How does the reflection principle apply in other geometric problems?
4. What if the swimmer had to make two stops on the beach? How would we approach that problem?
5. Can this problem be solved using calculus to verify the shortest distance?

Tip: The reflection principle is a powerful tool in geometry for solving problems involving shortest paths and minimizing distances.