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Old 2021-09-01, 15:15   #34
uau
 
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Quote:
Originally Posted by Dr Sardonicus View Post
As to proving the "right angles" idea correct (I'm not sure in what generality), I'm not sure how to do that short of invoking the "calculus of variations."
I think it should follow from a half circle being optimal in the straight-line case.

By zooming close enough to any smooth curve, you can approximate it arbitrarily closely by a straight line. So by looking at a small enough scale, you only need to consider how the fence can meet a straight line (plus corners). Now fix a point on the fence some distance away from the line, and ask whether you can improve the area between that point and the line for the same fence length. The question becomes "given a point a distance x from a line and available fence length c*x (c >= 1), maximize the area to a given side of the fence". Half-circle being optimal in the straight-line case implies that all its parts are optimal - for every distance from the straight line, for the existing arc length you can't do better than the circle arc that meets the line at a right angle.
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Old 2021-09-05, 00:11   #35
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So, I finally had some spare time for this.

FWIW:
The optimum distance between the 2 circles should be less than 1/2 meter away from 66.5 m.

I will attach the SolidWorks files next.
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Old 2021-09-05, 00:23   #36
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SolidWorks files are not valid for upload so they were zipped.
Attached Files
File Type: zip EDG-500-A.zip (153.0 KB, 17 views)
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Old 2021-09-05, 01:10   #37
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/gasp/ ...you couldn't just solve it analytically?

In Russian there is a saying, "To shoot sparrows with a cannon". That's what's going on here.
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Old 2021-09-05, 01:28   #38
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Well, I could probably write a PARI-GP code to do the same. But for me CAD is the Path-of Least-Resistance.
If you have something else in mind, then you are better in geometry than I am.
ETA It would probably take me a month to write an equivalent PARI code if at all.

Last fiddled with by a1call on 2021-09-05 at 02:01
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Old 2021-09-05, 07:48   #39
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Quote:
Originally Posted by Batalov View Post
/gasp/ ...you couldn't just solve it analytically?
Shhh! Hush! We actually like what a1call is doing here.

We ourselves solved few of the "ponder this" challenges with autocad too, in the past (here is the most "famous", with link back to the forum, the cad solution is in the last post in the thread).

There is this anecdote about Edison wanted to find out the inside volume of the light bulb, to know how much air must be taken off to reach the desired pressure inside. In the beginning, the light bulbs were not the pear-like format they have today, but more "twisted", according with the technology at the time, so he called few of his mathematicians (he had many employees already in his factories and labs, before making the bulb, albeit people associate him with the bulb), gave them a glass bulb and asked them to compute the volume. He left and came back after 20 minutes, finding the mathematicians arguing about the final solution, in front of few pages filled with integrals, he got really pissed off, took the light bulb, filled it with water from the sink, poured it into a graduated cylinder, read the measurement and left for good. It took him 15 seconds. The mathematicians probably are still arguing today, in some corner of the building, long beards, under neon tubes or LED lights...

Last fiddled with by LaurV on 2021-09-05 at 07:53
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Old 2021-09-05, 13:51   #40
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I have no objection to using graphical representations and bracketing to approximate the solution.

That said, as has already been indicated in this thread, there are fairly simple formulas for the quantities required, assuming the lake has radius 1. I'm a dunce at programming, but, with the formulas in hand, it took only a few minutes to write a Pari-GP script to print out the numbers. The biggest programming challenge was solving the equation resulting from the condition that the length of the fence is 4 (In the original formulation, the lake has radius 50 and the fence has length 200, 4 times the radius of the lake.) I resorted to Newton's method, and to program that I had to take some derivatives involving trig functions to get the appropriate formulas.

The biggest real-world challenge in implementing the program was tracking down and correcting the typos in my script
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Old 2021-09-05, 14:24   #41
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Suppose the farmer has to enclose the roof with part of the 200m roll of fencing and that roll is 1m wide (and that the wall is 50m high). He may, by doing some simple wood work, run two or more pieces of fencing in parallel and in any Cartesian direction. What is the maximum volume? Will he have any fencing left over?

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Old 2021-09-05, 18:31   #42
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Anyone have ideas for how to solve a 3D version? For example, if you have a unit square and 2 units of freely shapeable surface, what's the maximum volume you can enclose?
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Old 2021-09-05, 18:32   #43
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How many spherical cows can it enclose?
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Old 2021-09-05, 18:52   #44
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Quote:
Originally Posted by uau View Post
Anyone have ideas for how to solve a 3D version? For example, if you have a unit square and 2 units of freely shapeable surface, what's the maximum volume you can enclose?
If I understand correctly, you are describing a square base (area = 1 units square) with a free-formed dome (area = 2 units square). A sphere is the most optimum general minimized surface, but it has a circular cross-section. So the dome here can't be part of a sphere. furthermore the surface would not be constructible using arcs and lines which would make it next to impossible to parametrize. There might be a surface/volume formula, but It certainly beyond my know how, which by itself does not mean much at all.

ETA: This is somewhat relevant:

https://en.wikipedia.org/wiki/Minimal_surface

Last fiddled with by a1call on 2021-09-05 at 18:56
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