## Up And Down The Mountain

**One day at noon, Annie runs to the top of a mountain. She sits and ponders the meaning of life until the next day at noon at which time she runs down the mountain along the same trail that she ran up. Was she necessarily at some point on the mountain trail at the same time on both days? Prove your answer — Mr. Simmons**

## Your Thoughts…

wow…first to anwer-anyhoo-

I drew a picture for tI started to realize that speed really doesn't matter. If she went really slow then the point would be higher up, but if she went super fast the point would be lower.

This problem(kinda found it): I divided the trail into 4 parts, and at each point there is a different time. assume that the speed is 1mi./hour(the speed doesn't really matter, but for this picture, since there are different times, I have to assume a speed).

the pic is represented for something different, but it works…

So, from the picture, we know that the only time she is at the same point at the same time in the two days is at the midpoint( if the speed is the same; if it's not the same, then it will be somewhere on the trail). So, the answer is **YES**.

- Madhula A

Those are nice pictures, but I have no clue what to do. I think I agree.

- Josh F.

I agree with Madula about the times but the only thing not explained is if she runs down at the same speed as she does up. Most people move faster downwards so if she does move faster down then there is no time when they are the same.

Brad S. has spoken! Tee hee. :)

What about the peak? wasn't she there too … like if you think about it ? Because she could be at the peak … and the peaks a part of the trail too right?

♪Henry C ♫

Wait .. never mind… same time.. my bad i was wrong, .yea Madhula i think your right..

♪Henry C ♫

Necessarily means you have to. So, did annie have to at some point on the mountain trail be at the same time on both days? the answer is no. She starts at the bottom and goes to the top on the first day, and top to bottom on the second. So, if she were running at the exact same speed, the only point that she would reach at the same time would be the midpoint. But, this is nearly impossible for a human, and absolutely robotic. Think about it: running at the exact same speed? No me digas! Plus, she would down faster than she came up. And, she probably couldn't climb a mountain at a consistent speed. She would have to run some, and then stop, or run at one speed, then another. It would be impossible to time yourself accordingly to be at the same place at the same time on a different day. And, if it were somewhere else on the trail, then it would be at a different time. And, there is a way where you can purposely make it not the same time at the same place. Assuming she ran the exact same speed both ways, she could stop right before the midpoint while coming down, but keep on running on the way up. Then, they would meet at the same point, but at a different time. Think of it like this: at 1:30 p.m. on the way up, she is exactly at the midpoint. She continues running at the same speed on upward, neither accelerating nor deccelerating until she reaches the top. On the other hand, as she comes down, she stops right before the midpoint at 1:29, waits, and then crosses the midpoint at 1:31. The place where she had stoppped right before was crossed the previous day at 1:31, making no placed reached at the same time.

-**Jonathan Y.**

What if there was a trail for hiking there already and she paced her self ?

♪Henry C ♫

Well, this problem would be difficult to assume the variables. Unless you count that she runs the same pace and speed up and down the mountain, then it would be yes. If this was in real life, then it would be almost impossible. Plus, the problem doesn't even tell you what time she reached the mountain and what time she reached home. So I can't make up my mind…

But what I'm going is right now, is with Madhula, assuming that she runs the exact same pace up and down the mountain with some sort of complicated device determining her speed and distance from home and mountain, and with robotic legs adjusting to these variables. This would make her reach the same spot at the middle of both her runs. :)

—**John Y.**

i do not agree with madhula- unless she was running at a constant pace, then it is not nessecary (UNLESS THE STARTING AND ENDING COUNT AS POINTS ON THE TRAIL)

(REVERSE FOR DAY2) Day 1 Day 2

Point 1 12:00 12:00

Point 2 1:05 1:10

Point 3 1:55 2:00

WAIT A SECOND…, I JUST REALIZED SOMETHING!

Point 4

Point 5

Point 6

Point 7

PEAK

GRAPH IT!!!!!!!!!madhula's right!!!!!!!!!!

-rohan