The grazing Goat Problem

This is an old math problem which, I was told, is phrased in the following way :

A farmer has a circular pasture (circle 1 below) and a goat which he ties to a rope of given length ( = radius of circle 2 ). The rope then is tied to a stake. How far away from the center of the circular pasture must the stake be placed in order for the goat to be able to devour only exactly one half of the pasture before the stake is being moved. The numbers I entered into the boxes below equal those I were told by a friend of mine.

From the math point of view : it is possible to develop an equation which allows you to determine the size of the overlap region of two circles given the radii of both circles and the distance between their respective centers. But, you need to know your trig functions. Unfortunately, this equation is not invertible; you cannot solve it for the distance given the area.

The problem though can be solved to any desired accuracy by starting with a good estimate for the distance and calculate for that the size of the overlap region and derive from that a better approximation to the correct value of the distance. Repeat that until the changes from one cycle to the next are small enough. This is called an iterative procedure. Luck will have it that the Newton-Raphson method can be employed.

If you click on the button 'Click to determine distance' two tables will appear which for the values presently in the boxes present the size of the overlap region divided by the area of circle 1 ( column name α ) as function of the distance between the two centers (column labelled s). The values of s range from the case of the circle just touching each other ( α=0 ) to when either circle 1 is fully engulfed by circle 2 ( α=1 ) for the case of r2 > r1 or circle 2 just is completely inside circle 1 (but still touching each other ) in the case of r2 < r1. The case of r2 < r1 is of course undesirable from the farmers point of view because the goat cannot eat the left over area on the next day. But those are just details for the mathematician.

From the first table my program picks the distance which is closest and smaller to the desired size of the overlap region and then iterates to get a sequence of values for the distance, each having an overlap region whose size is closer to the desired value. This sequence is displayed in the second table.

The two tables are followed by a plot displaying the two circles according to your specifications. You may vary your input and click on the 'Click to determine distance' button as often as you like to satisfy your curiosity.

Enter radius of circle 1 (= radius of pasture) :

Enter radius of circle 2 (= length of rope) :

Enter fraction of area of overlap region to area of circle 1 :

Zig Herzog;
Last revised: 08/29/16