Goat in the field

Originally posted by PBE6
How did you get off your rope? 😕

I chewed it off with my butt.

Originally posted by PBE6
This is an old one that I posted a few years ago, but I don't remember it being answered analytically (not sure if it's possible or not). So here we go again!

A farmer owns a circular field, and wants to put his goat in it. However, he only wants to allow the goat to graze over half the field (i.e. he's a sadist). If the goat is tied to the edge of the ci ...[text shortened]... PS - This is easily solvable numerically, but can anyone come up with an analytical solution?

The required length (length of rope = L ) of the rope as a fraction of the diameter (diameter=D) is given by:-
L/D = cos (x/2) where x is the solution of the equation
sin x - x * cos x = pi/2 [ x in radian measure ].
And of course 'pi' is the ratio of the circumference of a circle to its diameter.
It turns out that x is only slightly greater than pi/2. and cos x/2 is only slightly less than sqrt(2).

Originally posted by ranjan sinha
The required length (length of rope = L ) of the rope as a fraction of the diameter (diameter=D) is given by:-
L/D = cos (x/2) where x is the solution of the equation
sin x - x * cos x = pi/2 [ x in radian measure ].
And of course 'pi' is the ratio of the circumference of a circle to its diameter.
It turns out that x is only slightly greater than pi/2. and cos x/2 is only slightly less than sqrt(2).

Yep, that's right. I just checked out this problem of wolfram.com again, and it says there's no analytical solution (only a numerical one, which you'd have to use to get a number out of ranjan's solution).

Originally posted by ranjan sinha
The required length (length of rope = L ) of the rope as a fraction of the diameter (diameter=D) is given by:-
L/D = cos (x/2) where x is the solution of the equation
sin x - x * cos x = pi/2 [ x in radian measure ].
And of course 'pi' is the ratio of the circumference of a circle to its diameter.
It turns out that x is only slightly greater than pi/2. and cos x/2 is only slightly less than sqrt(2).

In fact, cos x/2 is less than 1/sqrt(2). The solution for x obtained by numerical iterative methods comes to:-
x = 1.9056959293.
Accordingly L/D = 0.579364235, which is correct to 8 decimal places.
PBE6 is right. This problem cannot have an explicit analytical solution.

Originally posted by ranjan sinha
In fact, cos x/2 is less than 1/sqrt(2). The solution for x obtained by numerical iterative methods comes to:-
x = 1.9056959293.
Accordingly L/D = 0.579364235, which is correct to 8 decimal places.
PBE6 is right. This problem cannot have an explicit analytical solution.

That's the part that's really interesting to me! I realize that some equations can't be solved explicitly, but is there any way to prove that this one can't and will never be solved explicitly?

Originally posted by ranjan sinha
In fact, cos x/2 is less than 1/sqrt(2). The solution for x obtained by numerical iterative methods comes to:-
x = 1.9056959293.
Accordingly L/D = 0.579364235, which is correct to 8 decimal places.
PBE6 is right. This problem cannot have an explicit analytical solution.

So, in English then.

If the original circle is 1 metre in diameter, then the length of the rope must be 0.579 metres long in order cover 50% of the field?

Put another way, the rope must be 14% longer to cover 50% of the field.

Ok, I had 30% in my previous post but I think I forgot to divide by 2 to account for the lower half of the circle but at best, mine was just an estimate based on a2 +b2 = c2