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Sunday, November 15, 2009

Verdania: A Mathematical Odyssey: Chapter 7: Chasing

Chapter 7: Discovering
After Jeanie cracked the code to the stone door, Jesse, Miguel and the Captain pushed up against it and it swiveled open with remarkable ease. Cynthia stood to the left of the group and hugged Justin close to her. When the   Captain motioned for the group to follow her, Cynthia shook her head. “We don’t know what we are doing, Captain. I’m not going to risk my son’s life for this crazy chase.”
“Suit yourself, Cynthia. I’m not giving up on my nephew. Anyone else want to stay back? Go ahead.” She entered into the cave with Jesse by her side. Miguel looked at Marge and Jeanie, then they entered as well. Justin looked at his mother, perplexed. 
“Mom, we can’t just stay here and do nothing. Besides, there is probably more danger here than inside the cave. Let’s go.” Just as he finished, though, they both heard an ominous grinding sound as the stone door started to close. Cynthia looked at Justin, who was tugging at her sleeve. “Come on, let’s go with the group.” She relinquished and they both squeezed into the cave just before the entrance completely shut up.
Cynthia expected complete darkness, but was surprised by the amount of ambient light. The walls glowed a golden tone. She touched the stone. It was cool to the touch. Justin was walking ahead of her, calling for the Captain. She followed.
The group of seven walked in amazed silence. Justin traced his finger along the glowing walls. This was not a cave as much as a tunnel. It seemed handmade. Where could it be leading?
The Captain stopped and cocked her head to one side. “I think I hear foot steps. Maybe it’s Justin.” She started running. Jesse ran alongside her, while the rest of the group jogged behind. Jeanie and Marge ran with easy, while Miguel was struggling with a sore foot. Justin could have kept up with the Captain, but stayed back with his mother, who hadn’t run in many years and was huffing and puffing from the exertion.
The tunnel started to slope downwards. There was a slight breeze, reminding Justin of the breezes he felt when the subway trains would arrive to their stations in New York City. He realized that probably meant there was an exit up ahead. As he turned a corner, he saw the glimmer of light ahead. As he got closer, he saw the Captain and Jesse standing there. Then Miguel, Jeanie and Marge stopped with them. He ran up a little ahead of his mother to see why they had stopped.
The tunnel opened up into a crater like valley, surrounded by steep cliffs and full of lush, verdant vegetation, lakes and streams. There was a red stone road leading towards the center of this valley, where it entered into a small jungle about a kilometer and a half from where they stood. The air was misty and quite a bit warmer than it had been on the other side.
Cynthia was bent over, braced against her knees, catching her breath. Jeanie was alongside her doing the same. The Captain turned to the group and said “I’m pretty sure I saw someone go into that jungle from this path. They were pulling a cart of some sort. I’m going to run down there with Jesse. Miguel, you stay back and take care of them.” She pointed to Justin and the others.
“No way, we’re all going together.” said Miguel, looking at Marge. She nodded in agreement. Justin looked at his mother, not sure she could run fast enough. 
“Well, ok, we’re off, then.”
The Captain, Jesse and Miguel started running at a brisk pace. Jeanie and Marge were right behind them. Then came Justin and his mother. It became evident, though, that they were all running at different paces. After about 10 minutes of running, Justin saw the Captain enter the jungle with Jesse and Miguel. Jeanie and Marge were about 300 meters behind her. He figured he and mother were another 120 meters behind them. He was not tired in the least and his mother seemed to be holding up ok, all things considered. 
Your Task: Assuming that the three groups of people were running at a steady pace the entire distance from the tunnel to the jungle, how far behind will Justin and his mother be when Marge and Jeanie enter the jungle? (hint: it won’t be 120 meters) What were the jogging speeds of each of the three groups?

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