# This circles question set that is the bane of my existence right now

#### glewlwyd-gafaelfawr

##### New member
I'm having a lot of trouble with this question -- my mom completely lost her cool with me because I just couldn't get my head around it! So now I'm on my own until I can solve it, which I can't because I'm very much trash at math. Anyway, I have to get it done as quickly as I can otherwise my mom will really lose it.

Q: A farmer ties a horse to a building on a 50 foot lead. The building measures 20 feet by 20 feet (floor). What is the maximum area the horse can use for grazing? If there are regions you can't find the area of, provide as good an estimate as you can. Assume the horse is tied to a corner outside the building, cannot get in, and that the building is not grazing area. (Remember, this will be based on parts of circles, no other shapes...the horse's rope will only get shorter when he tries to go around the building...)

1. How much of the 50-foot circle can the horse reach without getting interrupted by the building? What is that area?
A: So, since the radius (the lead) is 50, the area of the whole circle is 7853.98? And since it's only the 50-foot circle, I gotta remove the quarter part, 7853.98/4 = 1963.495, so 7853.98 - 1963.495??? I'm dying. 5890.485. Is that right?

2. Assume the horse has grazed all of the grass in the area covered by #1 and continues on around the building. What is the new radius when the rope is interrupted by the building? What is that area covered using this new radius of rope before the rope is interrupted by the building again?
A: Ok, I have no idea. This might as well be Greek because that's as much sense as it makes to me. Can someone please explain how this works to me in simple terms, please? What do they mean by the radius when the rope is interrupted? I can't wrap my dumb head around this at all. 'Course, I have no idea about the rest of the questions either.

3. What if the horse had gone around the building the other way. What would the new radius have been when the rope was interrupted by the building? What is that area covered using this new radius of rope before the rope is interrupted by the building again?

4. The areas you found in 7 and 8 overlap each other. How much do they overlap? What *approximate* shape do they make? What is that area?

5. What is the total area the horse can graze using your calculations from #1-4?

Thanks a bunch!

#### Subhotosh Khan

##### Super Moderator
Staff member
I'm having a lot of trouble with this question -- my mom completely lost her cool with me because I just couldn't get my head around it! So now I'm on my own until I can solve it, which I can't because I'm very much trash at math. Anyway, I have to get it done as quickly as I can otherwise my mom will really lose it.

Q: A farmer ties a horse to a building on a 50 foot lead. The building measures 20 feet by 20 feet (floor). What is the maximum area the horse can use for grazing? If there are regions you can't find the area of, provide as good an estimate as you can. Assume the horse is tied to a corner outside the building, cannot get in, and that the building is not grazing area. (Remember, this will be based on parts of circles, no other shapes...the horse's rope will only get shorter when he tries to go around the building...)

View attachment 28408

1. How much of the 50-foot circle can the horse reach without getting interrupted by the building? What is that area?
A: So, since the radius (the lead) is 50, the area of the whole circle is 7853.98? And since it's only the 50-foot circle, I gotta remove the quarter part, 7853.98/4 = 1963.495, so 7853.98 - 1963.495??? I'm dying. 5890.485. Is that right?

2. Assume the horse has grazed all of the grass in the area covered by #1 and continues on around the building. What is the new radius when the rope is interrupted by the building? What is that area covered using this new radius of rope before the rope is interrupted by the building again?
A: Ok, I have no idea. This might as well be Greek because that's as much sense as it makes to me. Can someone please explain how this works to me in simple terms, please? What do they mean by the radius when the rope is interrupted? I can't wrap my dumb head around this at all. 'Course, I have no idea about the rest of the questions either.

3. What if the horse had gone around the building the other way. What would the new radius have been when the rope was interrupted by the building? What is that area covered using this new radius of rope before the rope is interrupted by the building again?

4. The areas you found in 7 and 8 overlap each other. How much do they overlap? What *approximate* shape do they make? What is that area?

5. What is the total area the horse can graze using your calculations from #1-4?

Thanks a bunch!
Anyway, I have to get it done as quickly as I can
Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

#### glewlwyd-gafaelfawr

##### New member
Uh, I just dunno what they mean by the second question. If I don't know about the second question I can't proceed to the other ones. I just want an explanation of what the question could possibly mean.

#### Dr.Peterson

##### Elite Member
1. How much of the 50-foot circle can the horse reach without getting interrupted by the building? What is that area?
A: So, since the radius (the lead) is 50, the area of the whole circle is 7853.98? And since it's only the 50-foot circle, I gotta remove the quarter part, 7853.98/4 = 1963.495, so 7853.98 - 1963.495??? I'm dying. 5890.485. Is that right?

2. Assume the horse has grazed all of the grass in the area covered by #1 and continues on around the building. What is the new radius when the rope is interrupted by the building? What is that area covered using this new radius of rope before the rope is interrupted by the building again?
A: Ok, I have no idea. This might as well be Greek because that's as much sense as it makes to me. Can someone please explain how this works to me in simple terms, please? What do they mean by the radius when the rope is interrupted? I can't wrap my dumb head around this at all. 'Course, I have no idea about the rest of the questions either.
The wording is unclear, so I don't blame you for being confused. "Interrupted" is not a standard geometrical term.

But I've done many questions like this, so I can guess what they mean. They are guiding you step by step through the process of solving the main question.

I am pretty sure that (1) means "without letting the rope bend around a corner"; so you are right that they want the area of the whole circle, minus 1/4 of it (that is, 3/4 of the area of the circle).

In (2), they are allowing the rope to bend once. This adds to the area of (1) another 1/4 circle with a smaller radius (the amount of rope remaining past the corner). In (3) you'll be adding another 1/4 circle, in the other direction.

One thing you might do if you have more questions is to make your own drawing, as suggested, and shade in (in different colors) the areas you understand each question to be asking about. Then show us that picture, so we can check your understanding.

#### Subhotosh Khan

##### Super Moderator
Staff member
The wording is unclear, so I don't blame you for being confused. "Interrupted" is not a standard geometrical term.

But I've done many questions like this, so I can guess what they mean. They are guiding you step by step through the process of solving the main question.

I am pretty sure that (1) means "without letting the rope bend around a corner"; so you are right that they want the area of the whole circle, minus 1/4 of it (that is, 3/4 of the area of the circle).

In (2), they are allowing the rope to bend once. This adds to the area of (1) another 1/4 circle with a smaller radius (the amount of rope remaining past the corner). In (3) you'll be adding another 1/4 circle, in the other direction.

One thing you might do if you have more questions is to make your own drawing, as suggested, and shade in (in different colors) the areas you understand each question to be asking about. Then show us that picture, so we can check your understanding.
And add some nomenclature to some of the "important" points, like shown below:

#### glewlwyd-gafaelfawr

##### New member
The wording is unclear, so I don't blame you for being confused. "Interrupted" is not a standard geometrical term.

But I've done many questions like this, so I can guess what they mean. They are guiding you step by step through the process of solving the main question.

I am pretty sure that (1) means "without letting the rope bend around a corner"; so you are right that they want the area of the whole circle, minus 1/4 of it (that is, 3/4 of the area of the circle).

In (2), they are allowing the rope to bend once. This adds to the area of (1) another 1/4 circle with a smaller radius (the amount of rope remaining past the corner). In (3) you'll be adding another 1/4 circle, in the other direction.

One thing you might do if you have more questions is to make your own drawing, as suggested, and shade in (in different colors) the areas you understand each question to be asking about. Then show us that picture, so we can check your understanding.
OK, I'm really bad at this, let me try to make sense of it from your explanation.
1) The area of the whole circle (from the radius of 50) is 7853 (about), and 1/4 of that is 1963, so I do 7853 - 1963 = 5890.
2) So I gotta figure out the area of the green circle and add that to the 5890? Is that it? Sorry if I sound mega retarded but how do I go about doing that? I feel like there's something major in the green circle that I'm missing, lol. Please give me some pointers with that, if you want. Thanks!

#### Subhotosh Khan

##### Super Moderator
Staff member
OK, I'm really bad at this, let me try to make sense of it from your explanation.
1) The area of the whole circle (from the radius of 50) is 7853 (about), and 1/4 of that is 1963, so I do 7853 - 1963 = 5890.
2) So I gotta figure out the area of the green circle and add that to the 5890? Is that it? Sorry if I sound mega retarded but how do I go about doing that? I feel like there's something major in the green circle that I'm missing, lol. Please give me some pointers with that, if you want. Thanks!
The green circle is a 1/4 circle whose radius is DR and center at D.

Similarly analyze the red circle centered at A.

#### Dr.Peterson

##### Elite Member
Well, the green part is 1/4 of a circle, but that's all there is so far. You don't yet need to do anything more with it. The tricky part comes in part 4, and there they are letting you just approximate the answer rather than having to calculate it exactly, which can be done but is a lot harder than the rest.

Here's the scale drawing they suggested you make:

You found the yellow; now you've found the green including the overlap. Then you'll find the orange, then the overlap and the total. Just keep plodding along.

#### glewlwyd-gafaelfawr

##### New member
Sorry, I'm really, really bad at math. Like, it's almost unbelievable.
Woah, so the radius of green is... 30? Since the lead is 50 and the building is 20. So the area of the green circle is 2827. Is it the same with the red? Thanks.

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#### glewlwyd-gafaelfawr

##### New member
Ohhh, but it's only a quarter of the whole circle, so x 1/4 as well. 706? Then add that to #1 and I get 6595! Neat, if it's correct. And for #3 it's the same in the other direction (I think), so I add another 706 to the prev. result? I get 7301.

Damn, not I gotta figure out the overlap, though. I'm still pretty retarded on that one. Is it just a 10 by 10 square or something like that?

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#### Dr.Peterson

##### Elite Member
Sorry, I'm really, really bad at math. Like, it's almost unbelievable.
Woah, so the radius of green is... 30? Since the lead is 50 and the building is 20. So the area of the green circle is 2827. Is it the same with the red? Thanks.
Yes; that's all correct ... except that the green part is not the whole circle.

You've almost got 1, 2, and 3; for 4, remember they are only asking for a rough approximation. So if you think math has to be exact, let go of that and just be as inexact as you want. What does that shape look sorta like ...?

#### glewlwyd-gafaelfawr

##### New member
Okay, I think I got it:
It could pass for a square or a circle. It's a quarter of a circle with a radius of 10, so it's 78, but since it's like, more than that, it has to be more than 78 but less than 100 (the square, 10 x 10), so it's gotta be between 85 and 95?

#### Dr.Peterson

##### Elite Member
Okay, I think I got it:
It could pass for a square or a circle. It's a quarter of a circle with a radius of 10, so it's 78, but since it's like, more than that, it has to be more than 78 but less than 100 (the square, 10 x 10), so it's gotta be between 85 and 95?
You're a better mathematician than they expect you to be!

They seem to be expecting you to say simply that it looks close to a square, and therefore approximate the area as 100; but you have put it within an interval, which is a better way to handle an estimate. Good job!

(An exact answer would require some trigonometry and several steps; many people would use calculus instead. I know nothing about what you have learned, but I'm guessing this problem is intended for students who don't know any of that.)

#### jonah2.0

##### Full Member
Beer induced ramblings follow.
As Dr.Peterson indicated, "An exact answer would require some trigonometry and several steps ...", you could take it a bit further if you like.

Reversing the image and placing the upper right corner of the square structure at the origin of the Cartesian coordinate plane, you should have the following image where the exact area you seek might be

[imath]\pi\cdot50^2\cdot\frac{3}{4}+\pi(50-20)^2\cdot\frac{\frac{\pi}{2}-tan^{-1}\frac{b}{20+a}}{2\pi}\cdot2+\frac{1}{2}\left(20\right)\left(b\right)\cdot2 [/imath]