Area of Squares

Bob · 2012-11-14 06:32:44

hi zee-f,

My first drawing shows a bird's eye view, looking down. The barn is 20 by 20. It doesn't matter how high it is since the goat cannot fly!

If the goat is tied at A, it can certainly get to A just by letting the rope go slack. If it heads in the direction E it will have to stop at E because the rope is now tight. If it tries to go to the left it will be forced to go no further from A than the edge of the circle.

So it could go around in a clockwise direction, along the circumference of the circle, through F then G all the way round to H. At any time on that journey around the circumference you could look at the rope and it would go straight back to A in a straight line (the radius).

Also, the goat could get to any point inside the circle (the yellow part) , simply by heading back towards A. letting the rope go slack.

Does that make sense so far?

Bob

ps. picture shows goat just around from point F. Hard to find a view of a barn from above. Found one with no roof. Why? I don't know.

zee-f · 2012-11-14 08:57:44

yup So its just talking about what happens to the rope when the goat goes around the barn but when I read the problem it doesn't make sense its annoying.

Last edited by zee-f (2012-11-14 10:06:44)

zee-f · 2012-11-14 10:02:05

How much of the 50 foot circle can the goat reach without getting interrupted by the barn? What is that area?

So this is saying that from F to H = 50 foot?

zee-f · 2012-11-14 10:06:10

So the radius is 25 ?

zee-f · 2012-11-14 10:08:38

So the area would be = PI * 25² = (625 PI)

Bob · 2012-11-14 10:11:21

hi zee-f

The rope is 50 feet. Is that a diameter or a radius?

And what fraction of a circle is yellow in my first diagram?

Bob

zee-f · 2012-11-14 10:26:19

OOh then it's the radius so the diameter is 100

the fraction is 3/4

Bob · 2012-11-14 10:31:16

OK. So you can work out the area of grass the goat can reach round to point H.

Now think about what happens if the goat tries to carry on round in a clockwise direction from point H.

Bob

zee-f · 2012-11-14 10:37:50

it would only go 1/4 if the it tries to carry on round in a clockwise direction from point H.

Bob · 2012-11-14 10:46:14

What happens to the rope?

zee-f · 2012-11-14 10:49:27

So its still tied to A right? and it goes from H to f ?

zee-f · 2012-11-14 10:53:32

the rope will be half the size 3/4 - 1/4 = 1/2

Bob · 2012-11-14 10:58:58

I don't think you're getting it yet.

The rope is 50 feet long.

It is fixed at point A

When the goat is at H the rope lies along the line ABH.

AB = 20 feet.

If the goat tries to go round more beyond H what happens to the rope?

See new diagram below; rope in red.

Bob

zee-f · 2012-11-14 11:10:38

it will get tight and then the poor goat will die

Bob · 2012-11-14 11:14:19

Oh that would be cruel. I think you may assume that the goat only pulls on the rope to the extent of 50 feet maximum.

Did you see I have added a diagram to my previous post.

The red line represents the rope. When at H the rope lies along the side AB of the barn.

Then the goat moves round a bit more.

So what happens?

Bob

zee-f · 2012-11-14 11:18:08

it gets shorter so if it goes around and stops at c it will be 30 then to D it will be 10

Bob · 2012-11-14 11:25:52

Got it !!!

That's what I was waiting for.

Here's a picture for you of the possible maximum distance from the point A.

Bob

But there is one last twist to the problem. Say if you feel ready for it.

zee-f · 2012-11-14 11:35:56

I think I am ready

zee-f · 2012-11-14 11:43:37

this problem is annoying I feel bad for that poor goat hhhhhh

Bob · 2012-11-14 11:46:13

OK. Here we go.

Below is the same picture with the areas the goat can reach shaded yellow.

You have got to calculaute the area of all those bits of circles.

I hope you can do that.

I would advise splitting into three bits (50 feet circle, then 30 feet, then 10 feet)

Now for the twist. The goat goes round as far as he can and comes to a halt halfway between C and D.

He can see a particularly juicy bit of grass at point X. But the rope is fully stretched around the barn.

But he can reach it. How?

Bob

zee-f · 2012-11-14 11:54:39

ammm I have no idea

he will stretch his foot and pull it hhhh "joking" he cant get it

zee-f · 2012-11-14 11:57:51

but what if he went the other way around he can reach it

Bob · 2012-11-14 12:03:24

Excellent. Now you're talking. I think the goat will love you. All you've got to do is persuade him to go back and round the other way.

Below is the final picture showing the boundary of reachable areas.

The two routes lead to overlapping areas which is why the question talks about estimating areas.

Would you be able to make your own drawing on graph paper and count little squares?

Bob

ps. I see I've posted a diagram with backward letters on it (revealing how I made it). I'll edit it in a moment. DONE

zee-f · 2012-11-14 12:08:32

yup I can do that I will count each square as 10

Bob · 2012-11-14 12:11:28

Make it 10 x 10 = 100 square feet.

Bob

I'm off to bed now. Post me your answer when you can and I'll look at it tomorrow.

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