17 Oct '07 02:16>
Originally posted by PBE6I chewed it off with my butt.
How did you get off your rope? 😕
Originally posted by PBE6The required length (length of rope = L ) of the rope as a fraction of the diameter (diameter=D) is given by:-
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?
Originally posted by ranjan sinhaYep, 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).
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 sinhaIn fact, cos x/2 is less than 1/sqrt(2). The solution for x obtained by numerical iterative methods comes to:-
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 sinhaThat'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?
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 sinhaSo, in English then.
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.