20210901, 15:15  #34  
Jan 2017
2^{3}×3×5 Posts 
Quote:
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". Halfcircle being optimal in the straightline 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. 

20210905, 00:11  #35 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3^{2}·239 Posts 
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. 
20210905, 00:23  #36 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3^{2}·239 Posts 
SolidWorks files are not valid for upload so they were zipped.

20210905, 01:10  #37 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
17×563 Posts 
/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. 
20210905, 01:28  #38 
"Rashid Naimi"
Oct 2015
Remote to Here/There
3^{2}·239 Posts 
Well, I could probably write a PARIGP code to do the same. But for me CAD is the Pathof LeastResistance.
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 20210905 at 02:01 
20210905, 07:48  #39 
Romulan Interpreter
Jun 2011
Thailand
2·3·7·233 Posts 
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 pearlike 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 20210905 at 07:53 
20210905, 13:51  #40 
Feb 2017
Nowhere
4,993 Posts 
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 PariGP 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 realworld challenge in implementing the program was tracking down and correcting the typos in my script 
20210905, 14:24  #41 
Sep 2002
Database er0rr
53×73 Posts 
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?
Last fiddled with by paulunderwood on 20210905 at 14:34 
20210905, 18:31  #42 
Jan 2017
78_{16} Posts 
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?

20210905, 18:32  #43 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10011100011011_{2} Posts 
How many spherical cows can it enclose?

20210905, 18:52  #44  
"Rashid Naimi"
Oct 2015
Remote to Here/There
4147_{8} Posts 
Quote:
ETA: This is somewhat relevant: https://en.wikipedia.org/wiki/Minimal_surface Last fiddled with by a1call on 20210905 at 18:56 

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