A horse is tied at a point on the circumference of a circular grass field of radius r, by a rope with length L.

What is the length of the rope, in terms of r, so the horse can eat exactly half of the grass field?

This problem is just another version of "The Goat Problem".

The answer to "The Goat Problem", as given by the Wolfram Mathworld website, can be obtained by using the equation for a circle-circle intersection with the length of the rope given as the radius of a circle to which the horse is tied.  By assuming the radius of the circular grass field is 1 unit, an equation can be given in terms of r (the length of the rope):

pi/2 = r²cos^-1(r/2) + cos^-1(1 - r²/2) - r/2*SQRT(4 - r²)

This equates to rope length of approximately 1.15872847 units. 

As the problem is in the category of Calculus, an approach other than presented on the referenced website is probably sought.  I shall leave that to another.

Edited on June 15, 2008, 9:11 pm

Posted by Dej Mar on 2008-06-15 15:51:24