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A goat is tied to the corner of a small house, with a 6 m long rope . The house is 3 m wide by 4 m long its rektanguler house. There is grass around the house. On how much property can the goat graze?

I got the answer 95m2 but that answer is wrong so i dont really know anymore

Sarah
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    Draw a picture. (Not of the goat!) – copper.hat May 24 '16 at 18:05
  • differential equation? anyway, as you were told in the other question you did, the title must be informative and you are suppose to include your effort in solving this problem – Riccardo May 24 '16 at 18:05
  • Perhaps a more specific title would help attract Readers? – hardmath May 24 '16 at 18:05
  • The house is 12 m wide? – copper.hat May 24 '16 at 18:05
  • @copper.hat I doubt it. That would make it trivial. – almagest May 24 '16 at 18:08
  • @almagest: It was a request for clarification... – copper.hat May 24 '16 at 18:11
  • Is the goat tied outside the house, or inside? –  May 24 '16 at 18:18
  • How exactly is the house shaped. You say "3 · 4 m wide." Do you mean by this that the house is rectangular with $90^\circ$ corners with two sides $3$ meters and the other two sides $4$ meters? As suggested, draw a picture, and realize that the reachable area is a union of semi circles and quarter circles. – JMoravitz May 24 '16 at 18:18
  • Interesting elementary problem! Hint: the range of the goat can be decomposed in three parts: 3/4 of a big circle 1/4 of a medium circle and 1/4 of a small circle. Think to the bending of the rope when the goat reach the far corners. – guestDiego May 24 '16 at 18:20
  • Okay im sorry if im unclear but the only reason i asked for help is because ive bean trying to find the answer to this question for 2 weeks i just really want to know. Ive asked my teachers and parents and no one could find the answer so please help just help me im only twelve and haven't learned about this yet but i really want to know the answer – Sarah May 24 '16 at 19:46
  • I'm curious how you got 100 m^2. – fleablood May 25 '16 at 05:27
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    Imagine that there were no house and the goat were staked to a point. Then the goat could eat in a circle with radius 6 m. So the goat could eat the area of the circle: $\pi r^2 = 36 \pi m^2$. Now imagine the house were huge. The goat can not eat in one quarter. So it can eat 3/4 of the circle. That's $27\pi m^2$. Okay, but the house is only 4 m long (small house) so the goat can go around that corner as though it were staked with a 2 meter rope. For an additional quarter circle. ..... – fleablood May 25 '16 at 05:36
  • .... the circle is $\pi r^2 = \pi 4 m^2$. And the goat eats 1/4 of it. That's $\pi m^2$. So add that to $27 \pi m^2$ to get $28 \pi m^2$. Now we have to do the goat eating around the other edge. That's as though it were stake 3 feet away and has 3 more feet of rope. So the area of that circle is $\pi 3^2 m^2 = 9 \pi m^2$ the goat can eat 1/4 of that so that's 2 1/2 $\pi m^2$. So add that to what we had. The goat eats 30 1/2 $\pi m^2$. Final answer. – fleablood May 25 '16 at 05:43

2 Answers2

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As everyone suggests, once you draw a picture, the answer will be obvious.

The area the goat can graze will split into 3 circular sectors,

  • $\frac34$ of a circle of radius $6$ on southern and western sides of the house.
  • $\frac14$ of a circle of radius $3 = 6-3$ on northern side.
  • $\frac14$ of a circle of radius $2 = 6-4$ on eastern side.

This means the total area is

$$\frac34\pi (6)^2 + \frac14\pi(3)^2 + \frac14\pi(2)^2 = \frac{121}{4}\pi \approx 95.03317777109125 \text{(in sq. meter)}$$

A goat tied to a corner of a house

achille hui
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Here are some hints.

First, draw a rectangle $3$ cm high and $4$ cm wide. It should look like Achille Hui's diagram.)

Now, draw a line $6$ cm long from the lower left corner into the yard.

Now look at the area that the goat can go without catching the rope on the upper left corner, or on the lower right corner. What shape is it? (Think $9$ slices of a $12$-slice pizza.) What is the area? (The area of a full circle is $\pi r^2$, with $\pi \approx 3.14$ and $r$ being the radius of the circle.)

Now, if the goat is really hungry, it can go around to the right side of the house, or around to the top side of the house. How much rope is free on the top? (Think $6$ m - $3$ m.) What is the area of the patch that the goat can reach around with this amount of rope?

What about the bottom part? (Think $6$ m - $4$ m.)

Can you take it from here?

John
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  • Im pretty sure what you wrote would make sense to alot of people but i dont understand (im only 12) – Sarah May 24 '16 at 19:38