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

Hunterhill
Hunterhill
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January 6th, 2019 at 11:06:08 PM permalink
Just a wild guess 20 meters.
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Wizard
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Wizard
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January 7th, 2019 at 7:59:44 AM permalink
Quote: ChesterDog

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!



Very good work there! I'm impressed. Your answer is indeed close. You certainly deserve some extra credit, but I'm still looking for an answer to at least six decimal places. Even I don't know an exact expression of the answer.

Don's 24-hour waiting period has lapsed so, Don, you're welcome to jump in and prove your dominance.

As long as CD has posted a close approximation, I'll post this graph I did what the curve looks like.

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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January 7th, 2019 at 5:20:00 PM permalink
I'll give this until tomorrow morning for a complete answer. Otherwise, I'll post a solution and ChesterDog will get the beer.
It's not whether you win or lose; it's whether or not you had a good bet.
CrystalMath
CrystalMath
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January 7th, 2019 at 8:57:38 PM permalink

I get 49.437553.

I did need to re-learn some stuff, such as finding the length of a curve.

Using the basic formula, I want to find the distance from center, where the height of the curve is 50.

The basic formula is y = a*cosh(x/a)-a+10. For y=50, solve for x: x=a*cosh-1((40+a)/a)

The second part is to find the length of the curve. It's too hard to type here, but the length of the curve is
2*a*sinh(x/a) = 2*a*sinh(cosh-1((40+a)/a)) when plugging in the first formula.
Using this, we can solve for a, when the length = 100.
I just put this formula in excel and used Goal Seek to find a. Turns out a = 11.25 (I think exactly).

Then find the distance from the center to one pole: x=11.25*cosh-1((40+11.25)/11.25) = 24.7187765. The total distance between the two poles is 2*24.7187765 = 49.437533.

I also read that a = (horizontal force applied to rope)/(weight of rope). In this case, I will call the weight 100 and each end point supports 50 of that. The horizontal force = 50 * the sin of the angle at the end point = 50/(dy/dx(a*cosh(x/a)-a+10) = 50/(a*sinh(x/a)). This is the same formula that I found when I calculated the length of the curve.

Thanks, Mike. This was fun and it made me re-learn some things.
I heart Crystal Math.
ChesterDog
ChesterDog
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January 7th, 2019 at 9:14:03 PM permalink

I agree with CrystalMath's 49.437553.

(45/2)*ArcSinh(40/9) = 49.437553


onenickelmiracle
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January 7th, 2019 at 10:40:27 PM permalink
If I were to draw something 5 units high, and have 1 unit underneath the rope, still don't think I could imagine the rope with such a steep decline.
Winter's hope for us, a sunny day fools us, dreams die with a lie. They tried to kill us, Jimmy. They did. They're dirty f**king cops! *photo is not of an AP. Goal April 1, survey says! No 93,
Wizard
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Wizard
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January 8th, 2019 at 8:09:14 AM permalink
Quote: CrystalMath


I get 49.437553.

I did need to re-learn some stuff, such as finding the length of a curve.

Using the basic formula, I want to find the distance from center, where the height of the curve is 50.

The basic formula is y = a*cosh(x/a)-a+10. For y=50, solve for x: x=a*cosh-1((40+a)/a)

The second part is to find the length of the curve. It's too hard to type here, but the length of the curve is
2*a*sinh(x/a) = 2*a*sinh(cosh-1((40+a)/a)) when plugging in the first formula.
Using this, we can solve for a, when the length = 100.
I just put this formula in excel and used Goal Seek to find a. Turns out a = 11.25 (I think exactly).

Then find the distance from the center to one pole: x=11.25*cosh-1((40+11.25)/11.25) = 24.7187765. The total distance between the two poles is 2*24.7187765 = 49.437533.

I also read that a = (horizontal force applied to rope)/(weight of rope). In this case, I will call the weight 100 and each end point supports 50 of that. The horizontal force = 50 * the sin of the angle at the end point = 50/(dy/dx(a*cosh(x/a)-a+10) = 50/(a*sinh(x/a)). This is the same formula that I found when I calculated the length of the curve.

Thanks, Mike. This was fun and it made me re-learn some things.



I agree!!! Very good work there. You and both ChesterDog have both well earned a beer.



Good idea about putting the answer in the form of sinh^-1(x).

Looking up the equation for sinh^-1(x), I find an exact expression of the answer is (45/2) × ln⁡〖( 40/9+ √(1+ 〖40/9〗^2 ))〗


Wiz full solution (PDF)
It's not whether you win or lose; it's whether or not you had a good bet.
ChesterDog
ChesterDog
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January 8th, 2019 at 9:39:15 AM permalink
Thanks!
I did not know that sinh-1=ln[x+(1+x2)1/2].

And it's amazing to me that your (45 ln(40/9+(1+(40/9)2 )1/2))/2 simplifies to 45ln3.
Wizard
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Wizard
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January 8th, 2019 at 1:45:33 PM permalink
Quote: ChesterDog

Thanks!

I did not know that sinh-1=ln[x+(1+x2)1/2].

And it's amazing to me that your (45 ln(40/9+(1+(40/9)2 )1/2))/2 simplifies to 45ln3.



Good catch! I didn't know it simplified to that. Very elegant. For what it's worth, I chose the pole and rope length arbitrarily.
It's not whether you win or lose; it's whether or not you had a good bet.
CrystalMath
CrystalMath
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January 8th, 2019 at 2:09:33 PM permalink
Quote: Wizard

Quote: ChesterDog

Thanks!

I did not know that sinh-1=ln[x+(1+x2)1/2].

And it's amazing to me that your (45 ln(40/9+(1+(40/9)2 )1/2))/2 simplifies to 45ln3.



Good catch! I didn't know it simplified to that. Very elegant. For what it's worth, I chose the pole and rope length arbitrarily.



That is crazy. All the stuff inside the ln evaluates to 9, so we have 45*ln(9)/2, and ln(9)/2 = ln(3).
I heart Crystal Math.

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