Finding the Optimal Leash Length for a Spherical Grazing Area

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Discussion Overview

The discussion revolves around determining the optimal length of a leash for an animal tethered to a spherical cage, such that it can graze exactly half of the volume of jelly within a spherical grazing area. The problem is an extension of a previous geometry challenge and explores both theoretical and mathematical aspects of the scenario.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Julian introduces the problem by comparing it to a previous geometry challenge involving a circular field and proposes a 3D version with a spherical grazing area.
  • One participant suggests that the length of the leash should be between the radius of the sphere and twice the radius, indicating a range rather than a specific value.
  • Another participant shares a personal anecdote about a previous math experience, hinting at the challenges of providing precise answers in mathematical contexts.
  • A different participant claims to have calculated the leash length to be 1/10 longer than the radius of the sphere, seeking validation for this result.
  • Another participant describes their approach to solving the problem using geometric reasoning and integral calculus, arriving at a numerical solution of approximately 1.2285, but questions the validity of their assumption regarding the radius being equal to 1.

Areas of Agreement / Disagreement

Participants express differing views on the correct length of the leash, with no consensus reached on a definitive answer. Several approaches and calculations are presented, but uncertainty remains regarding the assumptions made in the problem.

Contextual Notes

Participants note the dependence on the radius of the sphere and the implications of assuming a unit radius for calculations. The discussion highlights the complexity of the problem and the potential variations in answers based on different assumptions.

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

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