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Farmer Templeton's Goat


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#11 Grayven

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Posted 09 December 2008 - 07:47 PM

It is not the rope that impedes the goat, it is the silo. As the goat, on the end of its rope so as to stretch it taut, approaches the silo, the rope follows the curve of the silo and is no longer a straight line.

Oh right, so the ends of the arc would curve in ever so slightly where they meet the silo...
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#12 Jiminy Cricket

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Posted 09 December 2008 - 07:48 PM

Seems to me that this estimation was a bit too liberal with the assumptions.

Spoiler for Here's what I get


Spoiler for Now, if you want to get REALLY precise


Wow, I can't even balance my checkbook!
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#13 bonanova

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Posted 09 December 2008 - 07:49 PM

It is not the rope that impedes the goat, it is the silo. As the goat, on the end of its rope so as to stretch it taut, approaches the silo, the rope follows the curve of the silo and is no longer a straight line.

Thanks. You're absolutely right.


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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#14 lazboy

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Posted 09 December 2008 - 07:52 PM

Spoiler for Shoot, while I was a-doing the math someone beat me to this!

Spoiler for good call

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#15 Prof. Templeton

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Posted 09 December 2008 - 10:57 PM

Lazboy has got the answer I was looking for. Congrats. I also like his attempt at figuring the lost area due to the rope wrapping around the silo. If the arc AB is 12 meters then I figure the grazing circles radius diminishes to about 11.5 meters where it contacts the silo.
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#16 HoustonHokie

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Posted 10 December 2008 - 07:28 PM

Lazboy has got the answer I was looking for. Congrats. I also like his attempt at figuring the lost area due to the rope wrapping around the silo. If the arc AB is 12 meters then I figure the grazing circles radius diminishes to about 11.5 meters where it contacts the silo.

Spoiler for day late, 0.75 m2 short

Edited by HoustonHokie, 10 December 2008 - 07:32 PM.

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#17 lazboy

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Posted 11 December 2008 - 04:09 AM

Spoiler for day late, 0.75 m2 short

Wow, I guess your precision trumps mine. I knew it was a spiral shape, but didn't take the time to figure out how to draw a perfect spiral (let alone figure out what specific type of spiral it should be) in the software I was using, so I used an arc tangent to the cyan circle in your drawing with the same start and end points as your spiral. So it was merely another estimation and not the exact solution, but if your solution is completely accurate, my "precise" solution was only off by 0.2%. :)
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#18 bonanova

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Posted 11 December 2008 - 05:15 PM

Spoiler for day late, 0.75 m2 short

FWIW - It's the type of spiral whose radius increases linearly with angle.
The Archimedean spiral: r = a + bθ
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#19 HoustonHokie

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Posted 11 December 2008 - 07:47 PM

FWIW - It's the type of spiral whose radius increases linearly with angle.
The Archimedean spiral: r = a + bθ

Is it still an Archimedean spiral if the center of the spiral moves with the angle as well? The graphic shown on the linking page you sent shows a definite central point - for this problem, that center does not stay still, but keeps moving around the silo. I've been able to define the spiral parametrically, but don't know of a single function which could describe it.
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#20 bonanova

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Posted 11 December 2008 - 07:54 PM

Is it still an Archimedean spiral if the center of the spiral moves with the angle as well? The graphic shown on the linking page you sent shows a definite central point - for this problem, that center does not stay still, but keeps moving around the silo. I've been able to define the spiral parametrically, but don't know of a single function which could describe it.

Probably not.

But the idea of adding to the radius in proportion to the angle of wrap probably makes it the closest choice.
That is, it rules out a slew of them - logarithmic, most importantly.

The radius is measured from a moving point, as you point out.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell




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