Problem 4-5

First Look at Problem 4-5

Comments

For the beginning I just have it set up so that Alex has his paths connected to all three the vertices and the sum of the distance of those paths. Moving Alex around, it seems his total distance is smallest around the center of the circle (but we'll have to look more into this). For Beth, since the problem isn't specific on where exactly on the beach her path has to go to, right now Beth has many variables. Beth is a point that is movable, along with points A, B, and C. After exploring, I found that there are many different places where I see that value 6.93, but never any smaller. So Beth will need to be investigated more too.

Just looking at Beth

Comments on Beth

For now let's focus on Beth. If we think through what we know about shortest distance, we know that the shortest distance from Beth to the beach will be perpendicular to the beach. So since we are looking for the shortest distances, we will make all the paths for Beth perpendicular bisectors to the beaches. Notice that no matter where we go, as long as we stay in the triangle, the distance to the beaches is 6.93. Beth can be on one of the beaches or she could be at one of the vertices, or inland; it doesn't matter. Something I found interesting though is that her total distance is also the height of the triangle. If she decided to live at one of the vertices, then two of her paths go away (as she's already at those beaches by being at the vertex of those beaches), and then there is one path to the remaining beach. since the quickest route is perpendicular to that beach, this path makes the same line as the height of the triangle. So no matter where Beth decides to live, her total distance will always be the same as the height of the Triangle Island.

Looking at just Alex

Comments on Alex

Looking at Alex, it appeared his distance got smaller toward the center. So I made point O, which is the orthocenter of the triangle. Note that since this is an equilateral triangle, the orthocenter is the same as the centroid, incenter, and circumcenter. These are all different ways of finding the center through means of altitudes, perpendicular bisectors, etc. But in the equilateral triangle this is all the same dot. We notice that when Alex is directly over point O, the distance is at the smallest.

Finish

Final Notes

This is our final applet, so we can see both Alex and Beth. Alex is fixed, as he where the orthocenter of the triangle is. Beth is still movable as she can be at any spot in the triangle. Alex has more distance to travel then Beth, as the vertices are farther than the beach, but Beth can choose to live with Alex or not.