Verdania: A Mathematical Odyssey: Chapter 10: Revealing

   It was a vexing dilemma that faced the shipwrecked crew on this island they came to know as Verdania. They couldn’t get enough information to understand where they were, much less how to leave. They were living in a tree house village of young people with abundant food and a tranquil lifestyle. But they sensed some latent danger lurking.

It was Miguel who finally took action. “Look, we accomplish nothing by staying here and waiting,” he said quietly in the company of the Captain, Cynthina, Jeanie and Marge. Justin and Jake were off with Teena and the other kids. “While they are nice enough, it’s obvious that we don’t fit in. They don’t really know how to help us anyway. I say we offer to accompany them on the next trip they take to deliver those mangos. I would bet that the adults will be a little more helpful.”

Teena and Justin came up to them. They explained their idea to her and she agreed whole-heartedly. “But you know, Justin and Jake really should stay here with us. Youngsters don’t belong on the Rim.” This puzzled Miguel and caused a worried Cynthia, Justin’s mother to react.

“I can’t just leave my son behind!” She looked at the Captain with slight panic. 

The Captain nodded, then said, “Ok, this is what we’ll do. Miguel, go with Jesse, Jeanie and Marge to the Rim and find out all you can about this island and its inhabitants. Cynthia and I will stay back here with the boys and explore this valley. I’m betting there is a way to send a message back to us.” She looked at Teena but saw absolutely no reaction. Was she even listening?

The next day the four college students took off with the two oldest members of the village to deliver the mangos. It was a long and tedious journey because they had to leave caches of mangos along the route. Jeanie and Marge were sure there was a better way to do this, but it was just another mystery amongst many. It bothered Jeanie to no end, but amused Marge, who found these customs charming.

The entire trip, after all the backtracking, took them almost a week. They had brought their own food so as to not disrupt the mango delivery system (a system, according to Jeanie, that was pointless and inefficient). The valley floor was flat and the path they took perfectly worn with the three grooves for the cart. About 70 kilometers into the trip they passed a perfectly round and placid lake with huge orange fish swimming near the surface.

When they were finally approaching the Rim, there was a group of adults waiting for them by a huge stone monolith alongside a beautiful lake. Two women came running forth and embraced the two young villagers who had accompanied them. The villagers looked at each other, confused. They were swept away by the two excited women who were chatting as if they had always known these two.  The other adults gathered around the Miguel and crew. They looked happy and welcoming, sort of like Teena and Otto had when they had first arrived in their village, but quite a bit more interested in the details.

“Welcome to Verdania. We have heard of your rather dramatic entrance onto our island. I am Dula.” Dula explained that she was the leader of the group of adults living on the Rim of the Great Verdania Caldera Valley. 

“You know about us?” asked Jeanie, incredulous.

“Of course we do. Nothing enters or leaves Verdania without us knowing. This is a dangerous island, as your boy Jake would have found out if we hadn’t saved him from the Deesors.”

“Deesors?” 

“We have a lot to explain, I know. But first we need to get back to  Sentry Point One. Follow me.” They started up a steep, serpentine path along the cliff wall that enclosed the valley. The famous Rim was like the edge of  a huge saucer. As they climbed up the trail, the valley came into spectacular view. Dula walked effortless up the path, all the while explaining the layout of the Verdania Caldera Valley to the new arrivals.

“The circular Caldera Valley has a 100 km radius. The Youngsters’ village is exactly in the center.  Here at Sentry Point One we are at the northern most point of the Caldera Rim. Hope Lake, which you passed on your journey, is located due south 30 km from here. Starting at Hope Lake and going due west you would eventually reach the Sentry Point Two on western portion of the Rim. At that point, if you go due south for 15 km, you would reach the Adult Village.”

Jesse was perplexed. It seemed to him that just like the Youngsters, the Adults were going about their tasks in confusing and inefficient ways. “Why not just make a path along the circumference? Wouldn’t it be shorter?”

Dula looked at him equally perplexed. “My dear man, our route IS the shortest route. And besides, it’s far too dangerous to be moving along the Rim on a regular basis.” Just then a huge rock came flying through the air and hit Jesse on the side of the head. He crumbled to the ground. Danger was apparently an understatement on the Rim, thought Marge.

Your Task:  “As the crow flies”, just how far away is the Adult Village from Sentry Point One? How long would the route be if it included a walk along the Rim? How long is the route the Adults currently use?