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jpalmour@gmail.com
science forum beginner
Joined: 20 Jul 2006
Posts: 2
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 Posted: Thu Jul 20, 2006 8:26 pm Post subject:
Divine apparitions in the tethered goat problem?
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I was modeling the following problem using GSP, and what appears to be
an apparition of Mary cradling Jesus appeared in the picture below:
picture: http://jpalmour.googlepages.com/MaryFromTheTetheredGoat.jpg
video: http://jpalmour.googlepages.com/Mary.avi
A goat is tethered to a cylindrical silo of radius r with a tether of
length s. What is the area of the field that the goat can graze in?
My solution is very ugly. Does anyone have a nice solution to this
problem? |
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jt64@tele2.se
science forum beginner
Joined: 05 Jun 2006
Posts: 30
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| Posted: Thu Jul 20, 2006 8:57 pm Post subject:
Re: Divine apparitions in the tethered goat problem?
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Well the problem description isn't quite clear to me is it a fag goat
gracing in the silo, or just a plain goat?
I don't think Mary would help out the fag goat, but you never know.
JT
jpalmour@gmail.com skrev:
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jt64@tele2.se
science forum beginner
Joined: 05 Jun 2006
Posts: 30
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| Posted: Thu Jul 20, 2006 9:06 pm Post subject:
Re: Divine apparitions in the tethered goat problem?
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You see my beleive is that Mary help them who help themself through
troubled times and acknowledge her divine intervention through
apparation, anyone who spends time thinking about how make a plain goat
gracing the silo best way probably a good person at heart.
JT
jpalmour@gmail.com skrev:
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Robert B. Israel
science forum Guru
Joined: 24 Mar 2005
Posts: 2151
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| Posted: Thu Jul 20, 2006 9:16 pm Post subject:
Re: Divine apparitions in the tethered goat problem?
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In article <1153427163.593854.308400@b28g2000cwb.googlegroups.com>,
<jpalmour@gmail.com> wrote:
Let the silo be centred at the origin, and the goat tethered at
(r,0).
For the simplest case, let's suppose s <= pi r, so the goat can't
get to (-r,0).
The boundary of the grazing region consists of the semicircle
[x,y] = [r + s cos(t), s sin(t)] for -pi/2 <= t <= pi/2,
the parametric curve
[x,y] = [r cos(t) - (s - r t) sin(t), r sin(t) + (s - r t) cos(t)]
for 0 <= t <= s/r
and its reflection across the y axis,
and the boundary of the silo
[x,y] = [r cos(t), r sin(t)] for -s/r <= t <= s/r.
The area enclosed by a simple closed curve C (counterclockwise) can
be calculated using a line integral
A = int_C x dy
After some messy integration, I get
A = pi s^2/2 + s^3/(3 r)
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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jt64@tele2.se
science forum beginner
Joined: 05 Jun 2006
Posts: 30
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| Posted: Thu Jul 20, 2006 9:49 pm Post subject:
Re: Divine apparitions in the tethered goat problem?
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Have you seen movie good will hunting?
Robert Israel skrev:
| Quote: | In article <1153427163.593854.308400@b28g2000cwb.googlegroups.com>,
jpalmour@gmail.com> wrote:
I was modeling the following problem using GSP, and what appears to be
an apparition of Mary cradling Jesus appeared in the picture below:
picture: http://jpalmour.googlepages.com/MaryFromTheTetheredGoat.jpg
video: http://jpalmour.googlepages.com/Mary.avi
A goat is tethered to a cylindrical silo of radius r with a tether of
length s. What is the area of the field that the goat can graze in?
My solution is very ugly. Does anyone have a nice solution to this
problem?
Let the silo be centred at the origin, and the goat tethered at
(r,0).
For the simplest case, let's suppose s <= pi r, so the goat can't
get to (-r,0).
The boundary of the grazing region consists of the semicircle
[x,y] = [r + s cos(t), s sin(t)] for -pi/2 <= t <= pi/2,
the parametric curve
[x,y] = [r cos(t) - (s - r t) sin(t), r sin(t) + (s - r t) cos(t)]
for 0 <= t <= s/r
and its reflection across the y axis,
and the boundary of the silo
[x,y] = [r cos(t), r sin(t)] for -s/r <= t <= s/r.
The area enclosed by a simple closed curve C (counterclockwise) can
be calculated using a line integral
A = int_C x dy
After some messy integration, I get
A = pi s^2/2 + s^3/(3 r)
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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Virgil
science forum Guru
Joined: 24 Mar 2005
Posts: 5536
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| Posted: Thu Jul 20, 2006 11:12 pm Post subject:
Re: Divine apparitions in the tethered goat problem?
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In article <1153427163.593854.308400@b28g2000cwb.googlegroups.com>,
jpalmour@gmail.com wrote:
For L = length of rope, R = radius of silo and 0 < R <= pi*R, so the
rope goes no more than half way round the silo, my solution is
Area = pi^L^2/2 + L^3/(3*R).
For L > pi*R it does get quite ugly. |
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Ben Rudiak-Gould
science forum Guru
Joined: 04 May 2005
Posts: 382
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| Posted: Fri Jul 21, 2006 12:22 pm Post subject:
Re: Divine apparitions in the tethered goat problem?
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jpalmour@gmail.com wrote:
| Quote: | Does anyone have a nice solution to this problem?
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On a related note, does anyone know of a suitable award I could nominate
this subject line for?
-- Ben |
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