Finding the Optimal Leash Length for a Spherical Grazing Area

[JCienfuegos]

Hello everyone
This problem is similar to a problem that appeared sometime back on this website called "last geometry challenge, very difficult!"
The problem was this: There is a circular field of grass of radius r surrounded by a fence. If aa sheep is tethered to the fence, how long should its leash be so that it eats only 1/2 of the grass in the circle.
I propose making this into a 3D problem. Make the circle of grass a sphere of jelly surrounded by a spherical metal cage. How long should the leash be if an animal is tethered to the surrounding spherical metal cage so that it eats only 1/2 of the volume of jelly.
Ill post the answer in a couple of days if anyone is interested.
Julian

 

[256bits]

Since the volune of a sphere is
V = (4 pi ) )r^44 / 3

Then the answer is somewhere between r and 2r.
( a range is an answer ) :)

 

[JCienfuegos]

I tried using this trick in high school. The teacher said what is sin342 or something like that and I put "something between -1 and 1". I got that question wrong.

 

[256bits]

I now get the leash to be 1/10 longer than the radius of the sphere.
Is that correct?

 

[JCienfuegos]

Well, that is not the answer i got. Now that think about it, my solution may be wrong, but I cannot think why.
What I did was this:
I imagined the sphere to be resting onto of the x axis, directly on top of the origin. I then tied the leash to the origin. I found the equation of the sphere would be the circle x^2 + (y-r)^2 = r^2 rotated around the y axis. The volume swept out by the leash would be the circle x^2 + y^2 = l^2. I revolved the region bounded by these two curves around the y-axis and set it equal to 1/2 * 4/3 * pi r^3. I assumed the radius to equal 1, since I figured the units of its length wouldn't change the answer.
Then I set up some integrals, and solved. I am left with 8l^3-3l^4-8=0, which I solved numerically and I got 1.2285.
The trouble is, what is the radius isn't one? Then the ratio 1:1.2285 isn't the same. If it is okay for the radius to be simply 1, then the answer is good.