[bonanova]

A 1-acre field in the shape of a right triangle has posts at the middle of each side, with ropes just long enough to reach the two nearest corners.

Sheep are tethered to the posts on the two sides, of length x and y respectively, and a dog is tethered on the hypotenuse.

How many acres of grazing area outside the field do the sheep have to themselves?

1. 0 acres - the dog can get to it all
   
2. 1 acre
   
3. 2 acres
   
4. None of the above
   
5. Can't tell without knowing x and y.
   

[Guest]

The sheep are grazing outside?

The dog is thethered inside?

[Guest]

When you consider the circle that the dog will reach, the corners of a right angle square thats hypotenuse is diameter, acute angled corners of triangle will be just on the perimeter (the angle looks to diameter is right angle). Thus the full triangle will be in dogs area. Result:0 acre.

With math:

The triangle, whoose area is max. is equi. triangle, if such a triangle has an area of a²/2, the hemiarea of circle is a² x pi/4, thus the trinagle is inside the circle.

Edited January 2, 2009 by nobody

[Guest]

I notice that the problem asks how much area OUTSIDE the field the sheep have to them selves. I did a quick sketch and agree with nobody that they have no free area inside the field. However, each sheep has a crescent shaped free area beyond the field available to them. I have know idea how to calculate that yet.

maybe I'm way off base but here's my guess.

[Guest]

1 acre for the sheep that the dog cannot get to. This is assuming x=y and that the tethers are long enough such that the arc created is coincident with the points of the triangle. (I don't think the tether length was given in the problem though, so that would make the answer the last one).

[Guest]

I gave up on solving this symbolically. Trigonometry and I are no longer on friendly terms.

Since all the tether lenghts are given as equal to half their side, I plugged in x=y and got 1 acre of free space outside the field, same as importantmonkey.

However, when I tried 2x=y, I got slightly more than 1 acre of free space. Am I making a math error, or is the answer that you must know the side lengths?

[Guest]

A 1-acre field in the shape of a right triangle has posts at the middle of each side, with ropes just long enough to reach the two nearest corners.

So is that means the rope on the hypotenuse side can reach the corner of right square angle? because i'm sure that the right square angle is the nearest or equal of the other 2

[Guest]

A 1-acre field in the shape of a right triangle has posts at the middle of each side, with ropes just long enough to reach the two nearest corners.

Sheep are tethered to the posts on the two sides, of length x and y respectively, and a dog is tethered on the hypotenuse.

How many acres of grazing area outside the field do the sheep have to themselves?

1. 0 acres - the dog can get to it all
   
2. 1 acre
   
3. 2 acres
   
4. None of the above
   
5. Can't tell without knowing x and y.
   

Well, assuming that there is field outside of this triangle (You asked how many acres of grazing area OUTSIDE the field the sheep have to themselves), I would say B. 1 acre.

The dog's circle has the diameter of the hypotenuse, giving him full range of the triangle. One half of the total area of the circle is inside the triangle, the other half is outside. The cumulative area of the circles of the two sheep are equal to the area of the dog's circle:

x^2 + y^2 = z^2

(pi * x^2) + (pi * y^2) = (pi * z^2)

Due to positioning of the triangle midpoints, the circles rest in such a manner that the dog's circle covers half of each of the sheep's circles, meaning their total grazing area is equal to one half of their total cumulative area. The dog's area is two triangles (2 acres), the sheep's combined area is equal to the dog's area (2 acres), and one half of that area is covered by the dog. Therefore, the sheep have 1 acre of grazing area outside of the triangle.

Edited January 2, 2009 by Llam4

[Guest]

again, I'm not a math expert, but..

The area of the dog's circle will always be MORE THAN twice the area of the triangle. As x gets larger than y, the area of the cirlce gets larger and larger than the constant area of the triangle => 1 acre.

Also, while each sheep has a hemisphere of area oustide the triangle, the dog's circle has a segment overlapping into each sheep's area.

Each sheep is left with a crescent shaped free space, which must be less than half the area of their circle.

The question is, are they constant areas, for all values of x and y?

[Guest]

my final answer.

I redid the math for lots of triangles, and found that while one of the crescent shapes may get very large as the triangle gets "skinny", the sum of the crescents, less the area of the dog's overlapping segments of a circle, is always 1 acre, for a 1 acre triangle.