Goat in the field

This is an old one that I posted a few years ago, but I don't remember it being answered analytically (not sure if it's possible or not). So here we go again!

A farmer owns a circular field, and wants to put his goat in it. However, he only wants to allow the goat to graze over half the field (i.e. he's a sadist). If the goat is tied to the edge of the circular field, how long should the rope be to allow the goat to graze over half the field? Give your answer as a fraction of the diameter.

PS - This is easily solvable numerically, but can anyone come up with an analytical solution?

It all depends upon the strength and determination of the goat.
If we're talking about an extremely hungry goat then he'll strain
and pull and reach further than the rope.
It's nearly impossible to tell with a live creature, rather than if you're using --- let's say a
blade tied to the rope. You can't tell how much the goat is going to eat, how far it will crane it's neck and other variables that I can't even begin to fathom.
I'm not just trying to sound smart here... I think I have a valid point.

Goats are funny.

Originally posted by inthegreenroom
It all depends upon the strength and determination of the goat.
If we're talking about an extremely hungry goat then he'll strain
and pull and reach further than the rope.
It's nearly impossible to tell with a live creature, rather than if you're using --- let's say a
blade tied to the rope. You can't tell how much the goat is going to eat, how f ot just trying to sound smart here... I think I have a valid point.

Goats are funny.

YouTube

Edit: And you ARE trying too hard to be smart.

Originally posted by inthegreenroom
It all depends upon the strength and determination of the goat.
If we're talking about an extremely hungry goat then he'll strain
and pull and reach further than the rope.
It's nearly impossible to tell with a live creature, rather than if you're using --- let's say a
blade tied to the rope. You can't tell how much the goat is going to eat, how f ...[text shortened]... ot just trying to sound smart here... I think I have a valid point.

Goats are funny.

You'll get your coat? Right then. 😞

Originally posted by PBE6
You'll get your goat? Right then. 😞

Fedxi!

I'm just saying... it's impossible to tell.

It ceases to be an exact science when you're using an animal.

You should join Mensa. I mean, if you can.

Good times!

Originally posted by Palynka
http://www.youtube.com/watch?v=we9_CdNPuJg

Edit: And you ARE trying too hard to be smart.

That video was hilarious!

🙂

Originally posted by PBE6
This is an old one that I posted a few years ago, but I don't remember it being answered analytically (not sure if it's possible or not). So here we go again!

A farmer owns a circular field, and wants to put his goat in it. However, he only wants to allow the goat to graze over half the field (i.e. he's a sadist). If the goat is tied to the edge of the ci ...[text shortened]... PS - This is easily solvable numerically, but can anyone come up with an analytical solution?

(radius of farm - X)2 + (radius of farm - X )2 = (radius of farm)2

Originally posted by inthegreenroom
I'm just saying... it's impossible to tell.

It ceases to be an exact science when you're using an animal.

You should join Mensa. I mean, if you can.

Good times!

Ahh, right! I forgot, one always ends up with fractions when counting sheep. 🙄

Originally posted by uzless
(radius of farm - X)2 + (radius of farm - X )2 = (radius of farm)2

that's a ridiculous equation.

You end up with a radius + 0.3 units as your approximate answer for any given radius.

Hmm...

How dare you treat my people in this fashion!

Originally posted by AThousandYoung
How dare you treat my people in this fashion!

How did you get off your rope? 😕

Originally posted by uzless
that's a ridiculous equation.

You end up with a radius + 0.3 units as your approximate answer for any given radius.

Hmm...

Its actually about 0.7. Where do you get 0.3? did you check your work? that comes out to more like one tenth the area. Suppose a radius of ten units, squared, 100, times PI, 314 square units and change. So do 3, squared, 9 times PI, 28 square units, less than one tenth. Now try 0.7
7 units squared, 49, times PI, is 153 square units. Divided by 314, about half. Wait, I think I see what you are talking about, starting at 0.7 units, you add 0.3 and end up with 1 which would double I think. That is the opposite though, that gives the radius to double the area, we want the radius to half the area. I guess we are talking about the inverse of the same problem, just semantics.

This is a very good problem. Even with a Maths degree I'm not sure I can do it! (Didnt take the Geometry units!!!)

However I will attempt it from basic principles
(... it is half term next week!)

Originally posted by PBE6
This is an old one that I posted a few years ago, but I don't remember it being answered analytically (not sure if it's possible or not). So here we go again!

A farmer owns a circular field, and wants to put his goat in it. However, he only wants to allow the goat to graze over half the field (i.e. he's a sadist). If the goat is tied to the edge of the ci ...[text shortened]... PS - This is easily solvable numerically, but can anyone come up with an analytical solution?

The definitive answer:

YouTube

Granny.