The Circular Field

Blood On The Tracks
Posers and Puzzles 27 Oct '16 14:03
1. 27 Oct '16 14:03
A farmer owns a circular field which has a radius of 20m.

He wishes to tether a goat to a post on the edge of the field (ie, the circumference of the circle) with a rope so that the goat will have access to exactly half of the field.

How long should he make the rope?
2. apathist
looking for loot
28 Oct '16 11:49
I smell calculus.
3. venda
Dave
28 Oct '16 12:08
Originally posted by Blood On The Tracks
A farmer owns a circular field which has a radius of 20m.

He wishes to tether a goat to a post on the edge of the field (ie, the circumference of the circle) with a rope so that the goat will have access to exactly half of the field.

How long should he make the rope?
I've seen this before I think.
The divisions are going to be "half moon" shaped but my geometry knowledge is not good enough to apply to the problem
4. BongalloJoe
Not Gone Yet
28 Oct '16 12:38
Im probably not right, bu in my minds eye it looks correct.

If the radius is A and the circumference is B, then wouldnt you need to do A+.5A=Half the circle?
5. 28 Oct '16 12:54
Originally posted by Andrew Kern
Im probably not right, bu in my minds eye it looks correct.

If the radius is A and the circumference is B, then wouldnt you need to do A+.5A=Half the circle?
Hi

I think you are saying, for my 20m radius field, that it should be 30m rope?

Afraid that isnt correct ~ would be too long, letting goat eat much more than half
6. 28 Oct '16 17:25
14.14m
7. Ponderable
chemist
28 Oct '16 17:45
Originally posted by yelrambob
14.14m
14.14m would be the answer for a full circle inside the circle. (So put the post 14.14 m from the border to get the goat to eat up to the border.
8. 28 Oct '16 22:54
14.14 isn't correct, as ponderable says, that is the radius of a circle with half the area of the field

But because the goat is tethered on the circumference, it doesn't eat in a full circle, it is basically 2 segments stuck together back to back.
9. BongalloJoe
Not Gone Yet
29 Oct '16 00:37
I know! Building a fence.

ðŸ˜‰ ðŸ˜‰ ðŸ˜‰
10. venda
Dave
29 Oct '16 08:28
Goats will eat anything,so in the real world it'd probably chew thro' the rope!!
11. joe shmo
Strange Egg
31 Oct '16 00:384 edits
Originally posted by Blood On The Tracks
A farmer owns a circular field which has a radius of 20m.

He wishes to tether a goat to a post on the edge of the field (ie, the circumference of the circle) with a rope so that the goat will have access to exactly half of the field.

How long should he make the rope?
There probably is a cleaner way. ( to be honest I'm not going to find a numerical solution...because I'm lazy and I don't suppose you are after a numerical approximation, However, this should yield the answer if I followed through)

Let the radius of the field = R
Length of rope = L

The equation that needs to be solved:

1/2* Field Area = Area of Circular sector of radius "L" subtended by angle φ(L,R) + 2* Area Circular segment of Chord Length "L" subtended by angle ß(L,R)

1/2*π*R² = 1/2*arccos [L/( 2*R )]*L² + R²*( arcsin[ L/R*√(1 - ( L/( 2*R))² ) ] - L/R*√( 1 - ( L/( 2*R))² ))

Solve for L numerically ( maybe there is a compact analytic solution I'm missing?)
12. joe shmo
Strange Egg
01 Nov '16 12:141 edit
Originally posted by joe shmo
There probably is a cleaner way. ( to be honest I'm not going to find a numerical solution...because I'm lazy and I don't suppose you are after a numerical approximation, However, this should yield the answer if I followed through)

Let the radius of the field = R
Length of rope = L

The equation that needs to be solved:

1/2* Field Area = Area of Ci ...[text shortened]... 2*R))² ))

Solve for L numerically ( maybe there is a compact analytic solution I'm missing?)
Well, I went through with it. I found a small error in my equation when I tried to solve it.

Corrected version:

1/2*π*R² = arccos [L/( 2*R )]*L² + R²*( arcsin[ L/R*√(1 - ( L/( 2*R))² ) ] - L/R*√( 1 - ( L/( 2*R))² ))

13. 01 Nov '16 16:07
Nice work, Joe

23.17 it is

when I first met this, many years ago, I used Newton Raphson to hone in on the best angle

full solutions are available on the 'net'!
14. 23 Dec '16 00:092 edits
Never mind need to figure out curve.
15. BongalloJoe
Not Gone Yet
23 Dec '16 03:33
Originally posted by joe shmo
Well, I went through with it. I found a small error in my equation when I tried to solve it.

Corrected version:

1/2*π*R² = arccos [L/( 2*R )]*L² + R²*( arcsin[ L/R*√(1 - ( L/( 2*R))² ) ] - L/R*√( 1 - ( L/( 2*R))² ))