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Wizard
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Wizard
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March 13th, 2020 at 8:54:01 AM permalink
To all my self-quarantining members, here is yet another challenging math puzzle. This one I think is harder than average.

Cone Mountain is in the shape of a cone. The radius is 2 kilometers and the slant height is 6 kilometers. There is a road that ends at an observation point half way up the mountain. The observation point is directly half way between the start of the road and the summit. The road takes the shortest possible route while also making a complete revolution around the mountain.

Question 1: What is the full length of the road?
Question 2: What length of the road goes downhill (if going to the observation point)?

Beer to the first satisfactory answer and solution to question 2, subject to the usual rules:

  1. Please don't just plop a URL to a solution elsewhere until a winner here has been declared.
  2. All those who have won a beer previously are asked to not post answers or solutions for 24 after this posting. Past winners who must chime in early, may PM me.
  3. Beer to the first satisfactory answer and solution, subject to rule 2.
  4. Please put answers and solutions in spoiler tags.
Last edited by: Wizard on Mar 13, 2020
It's not whether you win or lose; it's whether or not you had a good bet.
unJon
unJon
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March 13th, 2020 at 9:04:29 AM permalink
Does the road make one revolution between the bottom of the mountain and (x) the observation point or (y) the summit?

Edit: and what’s the difference between the road and the trail?
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Wizard
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Wizard
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March 13th, 2020 at 10:53:41 AM permalink
Quote: unJon

Does the road make one revolution between the bottom of the mountain and (x) the observation point or (y) the summit?



The observation point. The road ends there.

Quote:

Edit: and what’s the difference between the road and the trail?



Sorry, where I said "trail" I should have said "road."
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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Wizard
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March 13th, 2020 at 8:54:46 PM permalink
I got a response from a "beer club" member that makes me think I haven't stated the question clearly.



The blue arrow represents the bathroom at the observation platform. The pink part of the road is behind the mountain.

Forgive the awful graphic, but I hope it represents the goal is the observation platform. The road does not go to the summit on the way and it makes one revolution only around the mountain.

I also realize that saying the path is "straight" may lead to some confusion, as the math is clearly on a curved cone. How shall I put it? I think this would be correct -- the path can be any shape that gets to the observation point and makes a complete revolution around the mountain.
It's not whether you win or lose; it's whether or not you had a good bet.
ssho88
ssho88
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March 13th, 2020 at 9:22:45 PM permalink
Quote: Wizard

I got a response from a "beer club" member that makes me think I haven't stated the question clearly.



The blue arrow represents the bathroom at the observation platform. The pink part of the road is behind the mountain.

Forgive the awful graphic, but I hope it represents the goal is the observation platform. The road does not go to the summit on the way and it makes one revolution only around the mountain.

I also realize that saying the path is "straight" may lead to some confusion, as the math is clearly on a curved cone. How shall I put it? I think this would be correct -- the path can be any shape that gets to the observation point and makes a complete revolution around the mountain.



Quite hard, is it necessary/possible to define the path into an math equations ? I guess it is about distance in 3 dimensional.
Wizard
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Wizard
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March 14th, 2020 at 5:17:54 AM permalink
Quote: ssho88

Quite hard, is it necessary/possible to define the path into an math equations ? I guess it is about distance in 3 dimensional.



I'd like to not give hints until after the the 24-hour Beer Club waiting period is over. I will say I was able to do the problem. The way I did it is not that complicated, but it may not be obvious to solve it the way I did.
Last edited by: Wizard on Mar 14, 2020
It's not whether you win or lose; it's whether or not you had a good bet.
ssho88
ssho88
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March 14th, 2020 at 6:12:02 AM permalink


I guess the shortest path should be :-

a) Go straight to the observation point, Distance A = 3, slant height !
b) Then turn one round horizontally, Distance B = 2*PI, Edited should be 2 * PI !

Total shortest distance = 2*PI + 3 = 9.2832km

Is it correct ? LOL

Last edited by: ssho88 on Mar 14, 2020
charliepatrick
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March 14th, 2020 at 8:18:13 AM permalink
Quote: ssho88



I guess the shortest path should be :-

a) Go straight to the observation point, Distance A = 10^0.5
b) Then turn one round horizontally, Distance B = PI

Total shortest distance = PI + 10^0.5

Is it correct ? LOL

I'm not sure I agree with your figures...
Walking directly up to the observation point is surely along half the side, so is 6/2 = 3Km.
Then walking horizontally around the circle, it now has a radius of 1, so is 2 Pi. This gives a total of about 9.14 Km.
Surely as you approach the observation point, say 50m before you get there it is better to head off to one side to join the horizontal path - i.e go along one side of the triangle rather than two.
rdw4potus
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March 14th, 2020 at 8:41:17 AM permalink
Any answer >9km is wrong, else you could walk to the summit, twirl around the peak, and walk back to the observation post.
"So as the clock ticked and the day passed, opportunity met preparation, and luck happened." - Maurice Clarett
ThatDonGuy
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March 14th, 2020 at 8:46:16 AM permalink
Quote: ssho88



I guess the shortest path should be :-

a) Go straight to the observation point, Distance A = 10^0.5
b) Then turn one round horizontally, Distance B = PI

Total shortest distance = PI + 10^0.5

Is it correct ? LOL


I don't think so - instead of a cone, use a cylinder with radius 1 where the point is distance D from the start:

The "straight up and then around" method would be D + 2 PI = sqrt(D2 + (2 PI)2 + 4 PI D)
However, if you roll the surface out to a rectangle, the shortest distance is sqrt(D2 + (2 PI)2)

Then again, I assumed the cone solution would be similar, and the "downhill distance" would be zero.


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