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A goat is tied to an external corner of a rectangular shed measuring 4 m by 6 m. If the goat’s rope is 8 m long, what is the total area, in square meters, in which the goat can graze?

Well, it seems like the goat can turn a full circle of radius 8 m, and a rectangular shed's diagonal is less than 8m (actually √52), and so shouldn't it be just 6 x 4 = 24 sq metre? The answer says it is 53 pi, and I have no clue why it is so or why my way of solving doesn't work.

Updated: Oh, and the only area given is that of the shed's. How can I know the full area in which the goat can actually graze on?

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    The goat is outside of the shed, so it wraps around the shed. – Cameron L. Williams Sep 26 '16 at 14:22
  • Related, with a circle instead of the rectangle: https://math.stackexchange.com/questions/928008 (related to the goat problem). – Watson Sep 26 '16 at 16:17
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    "What is the total area, in square meters, in which the goat can graze?" Infinite. Reasoning: the very first thing the goat is going to do is get bored, figure out that the reason it can't wander freely is because of the stupid rope and collar, and will then either A) slip its head out of the collar, or B) chew through the rope. We raise goats.They have the problem-solving abilities of a 7-year-old human. Seriously - you don't confine a goat - you just convince it that it will have more fun/be better fed/have life easier "inside" than "outside". If a goat wants out, it will find a way. – Bob Jarvis - Слава Україні Sep 26 '16 at 16:51
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    6 x 4 that's the area of the shed. The goat grazes outside of the shed, not inside – njzk2 Sep 26 '16 at 16:59
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    The key word that you seem to have skipped over (as did I, when I first read this) is external. – Brian Sep 26 '16 at 18:15
  • The size of the field as delimted by its containing fence. Goats are quite capable of gnawing through rope. – Joshua Sep 26 '16 at 19:35
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    @BobJarvis Are goats (and 7-year-old humans) capable of escaping orbit and exploring the whole universe as well? – Amani Kilumanga Sep 27 '16 at 05:42
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    "The goat is outside of the shed" In what world can a goat graze inside a shed? How can so many people miss this?? :p – Stijn de Witt Sep 27 '16 at 07:27
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    @Stijn the shed is a greenhouse. Grass grows inside. – Matt Samuel Sep 27 '16 at 10:54
  • @AmaniKilumanga - please...don't give them ideas... :-) – Bob Jarvis - Слава Україні Sep 27 '16 at 11:20
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    How long is the neck of the goat? It adds to the circles :) unless the rope is tied to its mouth, in which case eating in general might become problematic. – Konerak Sep 27 '16 at 11:42
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    @AmaniKilumanga As a simplifying assumption, we posit that the goat lives on an infinite, flat plane. ;-) – Kevin Sep 27 '16 at 15:16
  • Hint: the goat is outside the shed. Draw a picture approximately to scale and look at the pieces. – Ethan Bolker Sep 26 '16 at 14:21

2 Answers2

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Parts of three different circles. enter image description here

Seyed
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    Man, with that goat you have on the picture, you are definitely the winner of this post :D – Futurologist Sep 26 '16 at 15:58
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    You have a magically appearing $m^2$ in an equation. – Carsten S Sep 26 '16 at 16:06
  • Agree with Carsten, you should check your units. Other than that great answer :-). – user35915 Sep 26 '16 at 16:10
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    Don't know what @CarstenS means by magically appearing m2. The units are clearly in meters, so the area would be in squared meters. Perhaps he/she means that the m is missing and should be included in your previous terms. And +1 for the goat. – AgapwIesu Sep 26 '16 at 19:20
  • @amWhy,I myself didn't expect this much upvotes from these wonderful and kind people in here. – Seyed Sep 26 '16 at 20:29
  • @Thomas Oh, I know that. This was not a criticism of your post, you did fine. It's just sometimes, the way this site works, and responds, is inconsistent. That said, enjoy the ride! – amWhy Sep 26 '16 at 21:10
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    @AgapwIesu This is math.SE, so I'm thinking it's because meters weren't in the rest of the equation in this answer (even if they are in the question) – Izkata Sep 26 '16 at 21:42
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    Poor Thomas didn't get the full rep his goat drawing skills deserved. – Simply Beautiful Art Sep 27 '16 at 00:50
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    Wow nice answer. Clear case of 'a picture says more than a thousand words'. I was subtracting the shed area from the full cirkel area but your picture shows very clearly why that is wrong. Great answer, +1! – Stijn de Witt Sep 27 '16 at 07:30
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    @AgapwIesu: you can't write 53Pi = 166,5m²... your left hand would need m² as well :) – Konerak Sep 27 '16 at 11:39
  • @AgapwIesu, isn't that clear where that 53 comes from? Isn't it the result of mxm? – Seyed Sep 27 '16 at 13:30
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    A solution to the magically appearing square meters would be to stick them inside square brackets. Alternatively you can calculate everything with units written, but it gets unwieldy. Alternatively, I believe this notation is legit as well: $\frac{A}{m^2} = ...$ – John Dvorak Sep 27 '16 at 14:12
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    I'm going to wait until the UTC date turns to upvote this answer. You deserve every point. @AgapwIesu They mean the units should appear in all expressions of the equations, not just at the end. It's a great habit to be in if you do physics; it helps you spot mistakes. – jpmc26 Sep 27 '16 at 18:46
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    @JanDvorak yeah, but especially with that notation the $m^2$ looks like you have some variable $m$. Never set units in italic font; $\frac{A}{\mathrm{m^2}}$ is the correct style. – leftaroundabout Sep 27 '16 at 19:35
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    I am sorry, I had thought my current would be clear enough without extra explanation. The answer of course still fully deserves an up-goat. It is just that as a mathematician I take equality signs quite seriously and therefore flinch at $53\pi=166.5m^2$, because it would follow that $\pi=3.1415m^2$. On the other hand I would not mind the following: “$A=53\pi\approx166.5$. The area that the goat can graze is $166.5{\rm m}^2$.” Just my two cents. – Carsten S Sep 28 '16 at 07:03
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    @CarstenS I would prefer throwing the units in from the beginning. Dimensional analysis. – TRiG Sep 28 '16 at 11:29
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    Wow prettiest goat so far. +1 – mathreadler Sep 28 '16 at 19:57
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    For units, do not italicize. So $m^2$ is the square of something called $m$, but $\mathrm{m}^2$ means square meters. – GEdgar Sep 29 '16 at 12:44
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    Could we please keep the original goat? – camden_kid Sep 29 '16 at 14:15
  • @camden_kid, I would like to, but I am new in here and I really don't know how exactly the system works in here. I woke up in the morning and I saw my drawing was changed due to some other peoples decision. That interfering could be acceptable If there was anything wrong with my calculations or violating the rules of the site but not deciding to modify my cartoon drawing. I have the original drawing and can upload it again only if that is not against the "edit suggestion approved" rules in here. – Seyed Sep 29 '16 at 14:59
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    @Seyed Please upload it again and maybe put some text in the answer to tell people to stop changing the image. Great answer by the way. – camden_kid Sep 29 '16 at 15:07
  • @camden_kid, it is done. – Seyed Sep 29 '16 at 15:31
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    Two small points: You cannot "let" $\pi$ equal anything; $\pi$ is what it is. The answer is slightly wrong. The correct way to calculate is to use the full accuracy of your calculator (which knows $\pi$ pretty accurately), and round the result to the desired number of decimal places only when stating it at the end. – John Bentin Sep 29 '16 at 20:50
  • I've made this diagram into a graph where you can vary the side of the shed and the length of the rope. https://www.desmos.com/calculator/icntkxirrl – Jam Dec 16 '18 at 12:13
  • It has been a while and the goat ate a door in the center of the closest 6m side so now you have to recalculate. – Mark Schultheiss Sep 30 '19 at 20:23
  • This is just wrong as it currently is, the area isn't equal to 166.50. It's $53 \pi$, or approximately 166.5 [m^2] – ilkkachu Oct 01 '19 at 08:27
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$$\frac{3}{4}\pi 8^2+\frac{1}{4}\pi (8-6)^2+\frac{1}{4}\pi (8-4)^2=53\pi$$

Adi Dani
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