Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 261

Tethered Goat Problem

https://www.youtube.com/watch?v=QDnT4evo-5w&t=228s

(First question on this video)

Is the answer given correct? I don't get why the goat covers 3/4 of the large circle, but only 2 small fractions of the top left quarter.

I thought the goat could cover the whole circle, minus the area of (most of) the shed/barn (?). I thought it could go 10m in any direction from the bottom right of the shed.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,524

Re: Tethered Goat Problem

I've got a busy day so I won't be able to respond properly for a while.  A version of the tethered goat problem came up on the forum in 2015 and I made a series of pictures to show what's going on.  You won't find it in a search because this feature doesn't go back before the forum upgrade.

But you can find it by searching my posts.  Click on my name and then on show all posts.  There's a lot but you can home in on the right date.  I think my pictures are clearer than the ones on the vid.

The basic idea is that the goat can reach grass using the full extent of the rope until the rope catches a corner of the barn.  Then the available radius is reduced by the length of the side of the barn.

The measurements may be new. I'll try to look again tomorrow.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,524

Re: Tethered Goat Problem

I've found time to look at the video.  The measurements are different from the problem I met years ago and the working comes out more easily, so I thought it would be simplest to make a new picture.

If the barn wasn't there then the goat could reach all the grass inside a circle radius 10, so (pi times 10 times 10) in area.

But the barn gets in the way of the rope and there's no grass to eat inside the barn anyway. The orange area is the bit that the goat can reach without any rope snagging problems.  Hence 3/4 x pi x 10 squared.

When the goat is on the line AD produced, the point D acts as a new tether point with new radius 10 - 7.  Once again the goat cannot reach the whole of the new circle; only 1/4 of a circle radius 3.

Similarly when the goat is on the line AB produced the point B cats as a new tether point with new radius 10 - 9.  So the extra area here is 1/4 of a circle radius 1.

I agree with the video answer.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 261

Re: Tethered Goat Problem

Brilliant. Thanks a lot, Bob.

Of course, the barn gets in the way of rope. Doh!

Asking myself why I didn't see this. Could I have been thinking 'too mathematically'? I got caught up in pi, and radii, area, etc, that I forgot about the barn being a 3 dimensional object that is going to get in the way of the tether.

One thing about the video. The woman just said something like, "When the goat gets to here there is only 3 metres of rope left"; instead of saying what you said, i.e, here the rope gets caught on the corner of the barn. Maybe she thought it was obvious. And I see now that it was. Doh!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,524

Re: Tethered Goat Problem

It happens to us all. I wasted a large chunk of an exam trying to eliminate a variable and failing.  Afterwards I spoke to a friend who had done it with one simple thing that I'd missed. Ggggrrr!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 261

Re: Tethered Goat Problem

Yeah.