Any mathematicians out there??

I am stuck on the equations, calculus etc, but I can tell the farmer how to make sure the cow eats half the grass without doing any sums at all.

1) If he was the one who marked out that non-standard field, he should know where the centre is, as he must have used a post with a rope round it.

2) He could either: -
a) build a straight fence or wall across the circle going through the centre point, and untie the cow to eat the grass in the half she is loose in, or:-

b) do as in 3) to 7) (this is a much more elegant solution) :-

3) Mark a place on the each side of the circumference of the circle, directly opposite each other.

4) Using that same length of rope that he used to mark the original circular field, attach it to one of the marks, and draw a semicircle from the centre point of the circle to the circumference.

5) Attach the rope to the other mark and draw a matching semicircle from the centre point to the other side of the circumference.

6) Build a fence or wall along each semicircle, untie the cow, and put the cow into one of new fields.

7) He now has two Yin-yang shaped fields which are not only the same mathematical area but are also exactly balanced aesthetically and symbolically, so the chances of his cow eating exactly half the grass are three times as good as with just a boring straight fence.

Alternatively, if he doesn't have wood for a fence or stone for a wall, he could put the cow into the barn and wait until the grass has turned into hay, then feed half of it to the cow. Meanwhile it can eat silage from last year's grass crop, made more exciting by potato peelings and apple cores from the farm kitchen.

Posted one paragraph twice - if the farmer follows that suggestion twice he will have a very fat cow.

Keeper1

Get a smaller field

Buy another cow and let them share the grass?

I gave up and googled it. It's a long-standing mathematical problem, and even though I have a maths degree, I can't explain the answer!

He could divide the area into two circles, each half the same area, tether the cow in the centre with the rope at the correct length to wander around the inner circle. When the grass there is all eaten then lengthen the tether so the cow eats the grass in the outer circle.
Of course, the grass in the inner circle could start growing again so the cow could wander back and munch that as well.

Or just leave the cow to please herself and the farmer could go to the pub.

I spent a while establishing the equation to do just that (it is a long time since I did this kind of sum), before I realised that

a) just making the rope half the radius doesn't make an inner circle of the same area as the doughnut that is left. It is more complicated to work out how long the rope has to be to let Buttercup reach exactly half the area (but still possible)

b) more importantly, I think an important part of the puzzle is that she isn't tethered at the centre, but on the edge of the circle, so her "circle of influence" isn't even a whole circle. If you ask me, whoever invented this instrument of mental torture only did it to look down from a cloud and laugh at the poor saps trying to solve it.

At that point I decided to leave it to the experts and stay with lateral thinking and alternative solutions.

a) just making the rope half the radius doesn't make an inner circle of the same area as the doughnut that is left. It is more complicated to work out how long the rope has to be to let Buttercup reach exactly half the area (but still possible)

Yes, I realised that, that's why I didn't attempt to post an equation .

I do have a friend (ex-colleague) who just might be able to solve it mathematically but he's not a farmer so wouldn't be factoring in the cow and her moods.

Anyway, some of the grass might be longer than in other areas.
😀

It can’t eat exactly half if it is tethered on the edge of a circle, even if the rope is the length of the radius.

It doesn't give the length of the rope, so it could be longer than the radius The exact half doesn't need to end at a straight line, it could be a curve. If it goes over the diameter line at the centre, it could compensate by not stretching as far as the line at the edges.

Buttercup says she's thoroughly fed up, lonely and needs to be milked.
🐄

Grannynannywanny

You’ve got me stumped. I think I’d leave the cow in the adjoining field. Cut the grass on half the field, rake it up and throw it over the hedge 😀

I love your answer Grannynannywanny....

It is very cruel to tether the cow, so if I call the RSPCA they will take it away. End of problem.

He gets/borrows another cow. He tethers both to the stake in the middle of the circle of grass - and lets them sort it out. If they get their ropes tangled - then tough luck, they are after all imaginary cows.

If he tethers them to stakes opposite each other on the edge of the field, they will each eat a half-circle from where they are tied. Then he can move both their tethers to points at right angles to their first points, so that they can each eat the uneaten grass in the bits outside the semicircles they were eating first.

Maybe he can charge the rightful owner of the second cow for the grass that it has eaten, and use the money to pay for all the extra stakes and rope he has had to buy, and to pay the labourer with a hammer who banged the stakes in.

Just erect an electric fence across the middle, use a compass to find the middle. It's easy when you don't know how

I know of some cows that jumped over an electric fence and escaped.

Mogsmaw

I remember this from school. The maths teacher concluded it was a problem for advanced mathematics and not something we could solve at school. I suspected then, as I do now, that he hadn’t the skills to solve it!

I am a hearty fan of education, but honestly, some of the things we were taught ! Yours is a good example.

Mine is as follows: the whole class sent out to the pavement in front of the school armed with the blackboard ruler and protractor to calculate the distance the sun is from the earth.

Now why would even a maths mistress feel that every one of a class of 24 girls would either as schoolgirls or as adults find a use for this piece of information?

And if we did need it, I never have, we could have found the answer in any encyclopedia!

The rope should be one half of the diameter of the field divided by the square root of two - approximately 35% of the diameter.

And the application in the real world is….?

pen50 Sorry but I disagree with you. I think multiplying would work better than dividing, as the bit marked out by what the cow can reach while her tether is 35% of the diameter will be less than the 50% that she is supposed to be eating. I am trying to draw and post a diagram of what I mean but so far not succeeding.

Blinko

And the application in the real world is….?

There isn't one, Blinko It is purely to exercise the brain and keep it on its toes, just as going to the gym and doing repetitive movements has no application in the real world and is purely to exercise the body and keep it on its toes.

Goats can jump & climb & chew through ropes, so it must be a chain! Why would anyone have a circular field in the first place, or want it to eat only half the grass, or why they would want a crescent of grass left? Can't I simply move the stake to the middle of the circle & make the length of rope half the radius?

You and your brother need to find some other topics of conversation?!

Elegran

pen50 Sorry but I disagree with you. I think multiplying would work better than dividing, as the bit marked out by what the cow can reach while her tether is 35% of the diameter will be less than the 50% that she is supposed to be eating. I am trying to draw and post a diagram of what I mean but so far not succeeding.

I think you need to use sine because the area will be the same size as two segments of the circle. The area is relatively simple to calculate, but I can't remember much about trigonometry.

I can't remember much at all about any of the maths I was exposed to. At one time I could still prove with the appropriate diagrams that pythagoras theorem was true for any right-angled triangle, not just the easy 3-4-5 one and the other easy one (the sides of that now escape me, too, as well as the proof) I can still argue convincingly, given a long enough piece of string, that it is useful in the Real World if you are stranded on a desert island and want to build a house that doesn't have unsquare rooms.