# Very simple problem for my daughter but we, adults, find 2 different answers

I'm sorry I'm French so the subject may not be properly translated, but here's my try:

A goat lives in a rectangular place. She's tied to the point P. The length of the row is 8 meters. The problem is that she can eat flowers: it's the shaded area. The farmer doesn't want the goat to eat the flowers, so he has to strengthen the barrier. Draw the part of the fence he has to strengthen, and tell exactly how long this part will be. Measures are in meters.

I found out that I just have to draw one half circle, radius = $5$ meters (on the left), and another half radius = $3$ meters (on the right). But when calculating strengthened barrier I found different value from my wife's. How would you do?

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I think the issue is that the question is ambiguous as to what is allowed. – Charles Nov 3 '11 at 18:05
@Charles: I agree. I interpreted the strengthening to be the border of the shaded area. user9176 and you are building new fences in different locations. If that is acceptable, your solutions are shorter than mine. – Ross Millikan Nov 3 '11 at 18:08
@RossMillikan: Right. I was pointing out that even though yours looks worse than mine, it's optimal under the assumptions you used, just like I believe mine is optimal under the assumptions I used. Carl's solution is my favorite, though. :) – Charles Nov 3 '11 at 18:11

This assumes that the area to be strengthened is some of the border between the shaded area and the white area and no new fence may be built. The left extreme is 5 meters from the lower left corner of the hut. So it is 3 meters to the left of the left wall of the hut. The right extreme is 3 meters from the upper left corner of the hut, $\sqrt{3^2-2^2}=\sqrt{5}$ meters to the right of the left wall. The total length is then $3+\sqrt{5}$ meters.

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This is it. When drawing the two semi-circles, the length of the fence seems to be ~that. Thank you very much. – Olivier Pons Nov 3 '11 at 20:50

Wouldn't be extending the lower edge of the hut by $1+\epsilon$ be enough?

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the answer is clearly to make an infinitesimally small semicircle around P. – Carl Nov 3 '11 at 17:25
I've updated the question to be more "precise". Thank you for you answer, but this question is for an 11 years old child... so I bet she couldn't say that to her teacher ;). – Olivier Pons Nov 3 '11 at 20:49

Thinking like Mr. Goat :),

Approach A:

the only way he can reach the flowers is by:

(A) going around the hut (since it is a solid structure, and assuming he can't jump it) following the blue line until he reaches point F. This is 3+4=7 M.

(B) And from that point he could either go forward, right or left using the remaining length of the rope. This is 8-(3+4)=1 Meter. This length will be consumed either in the left direction or in the right direction so it will be used once only.

The total rope length is the lengths used in (A) + (B) which is 7 M.

The fence will be built at the red line with the length of 2 Meters, that is, 1 Meter in either directions.

This approach is simple but not quite correct (see comment below).

Approach B

This was added as a result of the valid comment below. The following argument is for the 2nd diagram.

1 - If the Goat moves from point P, the max. distance he could cover would be along the radius of circle A bounded by the rectangle sides, the hut walls and the horizontal line L since the roap will bend at the wall of the hut at point C. This path alone will not get the Goat to the flowers.

2 - If the Goat moves from point P along the side of the hut reaching point C, then the max. distance he could cover would be along the radius of circle B bounded by the hut wall. He may then go to corner point Q. The max. distance he could cover would be along the radius of circle C bounded by the rectangle sides and the hut walls.

The flowers will act as a tangent to circle C at 1 point (theoretically) and that is where the fence should be built.

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When he moves left or right, the rope will not stay fixed to the fence. It will move left or right with him. And, since the shortest distance between two points is a straight line, he will be able to move more than 1 m left or right, farther left than right. – Graphth Nov 3 '11 at 21:42
@Graphth, thanks for your valid point, I did not think of it. I tried to come up with another approach - I hope this one makes some sense! – NoChance Nov 3 '11 at 23:48

Place an $\varepsilon$-cm fence $\varepsilon$ cm to the left of P starting at the hut and extending downward. Then run $50+\varepsilon$ cm of fence from the bottom of the first fence toward the rightmost side of the pen. The goat's shortest path to the flowers is then $4.5+\sqrt{9+\varepsilon^2}+\sqrt{0.25+\varepsilon^2}\approx8+\frac76\varepsilon^2.$ This requires $0.5+2\varepsilon$ m of fencing for any $\varepsilon>0.$

If the goat has nonzero diameter you can set $\varepsilon=0$, nailing the rope to the hut essentially. If the fence has nonzero width you can actually get away with less than 0.5 m.

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If you go in diagonal down-right, you need a shorter fence... My intuition tells me that 45 degrees should be the best angle, but this can be calculated.... – N. S. Nov 3 '11 at 17:56
@user9176: Actually I think to the right is optimal. At a 45-degree angle you'd need > 0.57 m. – Charles Nov 3 '11 at 18:03
I've updated the question to be more "precise". Thank you for you answer. – Olivier Pons Nov 3 '11 at 20:48