Shortest+paths

The problem we asked was:


 * Some kids were hiking around and headed back to their camp at the end of the day. Sadly they needed to swing by the river to pick up some water for the camp. Naturally, they wanted to walk as little as possible, as they were all hiked out. What path should they take on their way back? **

Some of the questions that came up:
 * Is the camp by the river?
 * Is it on the same side of the river?
 * Is it on opposite sides?
 * What is the camp is between the hikers and the river (on a line perpendicular to the river from the hikers)
 * Is the river straight?
 * Are there more than one person present?

[|tc_1_4_26.pdf] [|tc_2_4_26.pdf]

There were several approaches offered, many ways of justifying that there is only one shortest path. It also took some convincing that the path produced was the shortest path. Other questions one might ask are: [|tc_3_4_26.pdf]
 * How will you locate the place on the river where you should head to (in reality)?
 * Can you use coordinates to find the path? Or to locate the point?
 * What if the river is circular? Or a parabola, or an ellipse?
 * Can you find a path whose portion from the hiker will make the angle with river twice as big as its portion from the tent makes with the river? -- this we worked on a bit.

We thought we were unsuccessful, but we just needed to connect some dots:

[|Lauren & Peter had the solution.pdf]

We moved on to:


 * Two cities are some distance apart and separated by a river. Choose a path there the road ought to be built. **

The conversation here also entailed first resolving some questions about what characteristics this highway ought to have and how we are allowed to build. Then we had come up with two solutions, which of course needed to be compared.

Should we post solution? Do you have questions? There are some interesting approaches...