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jpalmour@gmail.com
science forum beginner

Joined: 20 Jul 2006
Posts: 2

Posted: Thu Jul 20, 2006 8:26 pm    Post subject: Divine apparitions in the tethered goat problem?

I was modeling the following problem using GSP, and what appears to be
an apparition of Mary cradling Jesus appeared in the picture below:

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?
jt64@tele2.se
science forum beginner

Joined: 05 Jun 2006
Posts: 30

Posted: Thu Jul 20, 2006 8:57 pm    Post subject: Re: Divine apparitions in the tethered goat problem?

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:

 Quote: 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?
jt64@tele2.se
science forum beginner

Joined: 05 Jun 2006
Posts: 30

Posted: Thu Jul 20, 2006 9:06 pm    Post subject: Re: Divine apparitions in the tethered goat problem?

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:

 Quote: 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?
Robert B. Israel
science forum Guru

Joined: 24 Mar 2005
Posts: 2151

Posted: Thu Jul 20, 2006 9:16 pm    Post subject: Re: Divine apparitions in the tethered goat problem?

<jpalmour@gmail.com> wrote:
 Quote: 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
jt64@tele2.se
science forum beginner

Joined: 05 Jun 2006
Posts: 30

Posted: Thu Jul 20, 2006 9:49 pm    Post subject: Re: Divine apparitions in the tethered goat problem?

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
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Thu Jul 20, 2006 11:12 pm    Post subject: Re: Divine apparitions in the tethered goat problem?

jpalmour@gmail.com wrote:

 Quote: 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?

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.
Ben Rudiak-Gould
science forum Guru

Joined: 04 May 2005
Posts: 382

Posted: Fri Jul 21, 2006 12:22 pm    Post subject: Re: Divine apparitions in the tethered goat problem?

jpalmour@gmail.com wrote:
 Quote: Does anyone have a nice solution to this problem?

On a related note, does anyone know of a suitable award I could nominate
this subject line for?

-- Ben

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