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?

as stated by Dej Mar the Wolfram website gives the equation for the area of the intersection of two circles.

if the radius of the 2 circles are r1,r2 and the centers are distance d apart then the area is

sqrt[(-d+r1-r2)(-d-r1+r2)(-d+r1+r2)(d+r1+r2)]/d

now for our problem we make the following substitutions r1=L r2=r and d=r and thus the equation is simplified to

sqrt[L^2(4r^2-L^2)]/r

now to solve for the required L we equate this to half the cirlces area or pi*r^2/2

sqrt[L^2(4r^2-L^2)]/r=pi*r^2/2

2sqrt[L^2(4r^2-L^2)]=pi*r^3

4L^2(4r^2-L^2)=pi^2*r^6

16r^2L^2-4L^4=pi^2*r^6

4L^4-16r^2L^2+pi^2*r^6=0

setting x=L^2 we can use the quadratic equation to solve for x

x=(16r^2+-sqrt[256r^4-16pi^2r^6])/8

x=(16r^2+-8r^2sqrt[16-pi^2r^2])/8

x=2r^2+-r^2*sqrt[16-pi^2r^2]

but L can't be greater than 2r or the area covered would be the entire circle, thus we have

x=2r^2-r^2sqrt[16-pi^2r^2]

thus

L=sqrt[2r^2-r^2sqrt[16-pi^2r^2]]

for r=1 we have

L=0.872937

Posted by Daniel on 2008-06-15 21:06:19