Poll

5 votes (26.31%)
2 votes (10.52%)
1 vote (5.26%)
3 votes (15.78%)
7 votes (36.84%)
4 votes (21.05%)
2 votes (10.52%)
5 votes (26.31%)
2 votes (10.52%)
4 votes (21.05%)

19 members have voted

Wizard
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Wizard
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January 6th, 2019 at 5:29:00 PM permalink
I'm proud to say I have stumped the forum, apparently, except for Don, who may jump in in 3.5 hours.
It's not whether you win or lose; it's whether or not you had a good bet.
teliot
teliot
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January 6th, 2019 at 5:50:34 PM permalink
Quote: Wizard

I'm proud to say I have stumped the forum, apparently, except for Don, who may jump in in 3.5 hours.

I used to assign questions like this when I taught diff-eq, In my opinion, it's a bit advanced and not intrinsically interesting. Maybe that's the issue more than stumping folks. The one before this involved Riemann surfaces for the log function. Maybe you're missing your college days ...

I think source material from elementary number theory, geometry and recreational mathematics is more in the spirit of such questions.
Ace2
Ace2
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January 6th, 2019 at 7:04:11 PM permalink
Iím not familiar with catenaries or their properties but I had a look at the problem. My answer must be wrong since itís way too simple:

For the line which is the rope, x=0 being the center...

y = a cosh (x/a)
y = 50
a = 10

So x = 22.924 which is the x coordinate of the right pole. The total distance between would be double that or 45,85.

That could also be approximated by:

Ln (50 / 10 * 2) * 10 * 2 = 46.05
Last edited by: Ace2 on Jan 6, 2019
Wizard
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Wizard
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January 6th, 2019 at 7:05:19 PM permalink
Quote: teliot

I think source material from elementary number theory, geometry and recreational mathematics is more in the spirit of such questions.



Sometimes I take pleasure in a problem that helps me dust off some long-forgotten skills.

Anyway, if you care to suggest the next problem, I'm sure the forum would be up to the challenge.
It's not whether you win or lose; it's whether or not you had a good bet.
gordonm888
gordonm888
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January 6th, 2019 at 8:19:28 PM permalink
I just saw this a little while ago and I have struggled with this. Not sure this is the right answer.

Let 2X = the distance between the two poles so that the tops of the poles are are at (X,50) and (-X, 50). But placing the lowest point of the drooping rope at (0,0) the tops of the two poles are at (X,40) and (-X,40).

For a catenary, I think:
y = cosh (x/a)-a = 40
and the arc length is = sinh (x/a) which must equal 50 meters, half of the rope length.

so:

cosh(x/a) = (40/a)-a
sinh(x/a) = 50/a

Using Wizard's hint that cosh^2 (x/a) -sinh^2 (x/a) = 1

I get ((40+a)^2 )/a^2 - (50/a)^2=1

So a^2 = a^2 +40a+1600-2500
900= 40a

thus,
a=22.5

Thus sinh(x/22.5) = 50/22.5

so the half-distance between the poles, x = 22.5* arcsinh(50/22.5)
And the distance between the poles =2x= 45*arcsinh(50/22.5) which is approximately 69.2468 meters
Wizard
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Wizard
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January 6th, 2019 at 8:34:17 PM permalink
Quote: gordonm888

I just saw this a little while ago and I have struggled with this. Not sure this is the right answer.




It isn't what I get. Let's say there was a short pole 10 meters high directly between the poles. If the rope went from the left pole, directly to the top of the center pole and up to the right, wouldn't it's length be 2*((69.2468/2)^2 + 40^2)^0.5 = 105.9642 meters. We're given it is only 100 meters, so I don't think the poles could be that far apart.
It's not whether you win or lose; it's whether or not you had a good bet.
teliot
teliot
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January 6th, 2019 at 8:34:58 PM permalink
Quote: Wizard

Anyway, if you care to suggest the next problem, I'm sure the forum would be up to the challenge.

You know, I enjoy reading old math books as well. Over the last few days I re-read "Elementary Number Theory" by Burton. My goal was was to remember the easy way to prove the "Four Square Theorem" using the Quaternions, but it didn't come back. Sigh ...

This is actually a very easy read ...

http://www.mathcs.duq.edu/~haensch/411Materials/Quaternions.pdf
Wizard
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Wizard
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January 6th, 2019 at 8:42:04 PM permalink
Quote: teliot

You know, I enjoy reading old math books as well. Over the last few days I re-read "Elementary Number Theory" by Burton. My goal was was to remember the easy way to prove the "Four Square Theorem" using the Quaternions, but it didn't come back. Sigh ...

This is actually a very easy read ...

http://www.mathcs.duq.edu/~haensch/411Materials/Quaternions.pdf



I think I regret I asked. That is beyond my level. Pass the dunce cap, I'll be sitting in the corner.

p.s. Are Quaternions those things programmers use for moving graphics? I hope Ahigh sees this.
It's not whether you win or lose; it's whether or not you had a good bet.
teliot
teliot
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January 6th, 2019 at 8:54:34 PM permalink
Quote: Wizard

I think I regret I asked. That is beyond my level. Pass the dunce cap, I'll be sitting in the corner.

p.s. Are Quaternions those things programmers use for moving graphics? I hope Ahigh sees this.

No dunce cap for you!

You know the Quaternions, i, j, k and such ... Diana said she used the Quaternions in one of the slots she programmed -- she was very proud of it. I never saw her actual implementation, but you know she had some tricks.

A long time ago I sent you a book, The Incredible Dr. Matrix. Maybe try and find that and re-read a bit of it? I would say it was one of the best math books ever. There are some wonderful tidbits in there that might inspire some questions.
ChesterDog
ChesterDog
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January 6th, 2019 at 10:38:31 PM permalink
Quote: Wizard

A 100 meter rope is suspended from the top of two 50-meter poles. The lowest point of the rope is 10 meters from the ground. How far apart are the poles?...



I cut a piece of thread and taped its ends to the ceiling so that 100 cm of it were between the pieces of tape. Adjusting the tape so the thread hung down 40 cm, I found that the ends were about 50.5 cm apart. So, the answer to your question should be near 50 meters.


I tried to solve the problem with math, but eventually had to give up and learn the math from YouTube. It was fun, though--thanks for the problem!

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