20210829, 06:02  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
3·17·19 Posts 
Farmer Fred's funky field
Hi all,
Give this maximization exercise a try. Let Farmer Fred's field have a 200 meter fence. And more, the field is in the shape of a rectangle. There is a river on the East side of his field. There are right angles at each corner of this field. What dimensions should Fred choose for the 3 sides of his field in order to maximize the area of that field? Solution attached. Matt 
20210829, 06:15  #2  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2×23×137 Posts 
Quote:
a rectangle is a quadrilateral with four right angles Do you mean a triangle? 

20210829, 06:56  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2×1,087 Posts 
That is cool Matt. The fence being on only 3 sides, somehow changes the expected maximization (if it were on all 4 sides) at a square configuration to a rectangular one.
I haven't quite got my head wrapped around the mechanics of it yet. Something to think about overnight. Last fiddled with by a1call on 20210829 at 06:58 
20210829, 07:34  #4  
Jun 2003
3×1,733 Posts 
Quote:
EDIT: Let a,b,a be the three sides 2a+b=200 Maximize Area = ab = a*(2002a) = 200a  2a^2 Maxima will be when d(Area)/da = 0 i.e 200  4a = 0 => a = 50 So we need a 50/100/50 fence with area = 5000m^2 Last fiddled with by axn on 20210829 at 07:40 

20210830, 01:39  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
3·17·19 Posts 
Hi again all,
Hopefully this classifies this exercise. I want to make it clear to everybody. 
20210830, 02:23  #6 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3×5^{2}×79 Posts 
Most livestock will not respect a waterway as a fence. (Not that they show much respect for fences either. They need to be barbed wire, electrified, build to stop a light pickup truck, or all the preceding. Some farmers use railroad ties as posts.) They'll go down to the water to drink, and in doing so churn the bank into a muddy eroding mess. The EPA or state DNRs typically require fencing animals away from the water's edge, and maintaining a grass covered berm to prevent manure runoff from entering the waterway. The river edge costs MORE, not less, than landlocked fence lines. (edit) The berm is required even if livestock are not present or planned. 50 m x100 m = 5000 sq meters = 1.24 acres. That would be a lot of lawn to mow.
Last fiddled with by kriesel on 20210830 at 03:00 
20210830, 02:48  #7 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
14236_{8} Posts 

20210830, 03:51  #8 
Romulan Interpreter
"name field"
Jun 2011
Thailand
2650_{16} Posts 
Bwaaa haha! (sorry, I could not stop it!) Last fiddled with by LaurV on 20210830 at 03:52 
20210830, 04:04  #9 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2·23·137 Posts 
There are right angles, there are wrong angles, and there are triangles.
To make the puzzle more interesting allow the fence to be any shape of your choosing, and change the river into a lake of radius 50m. To maximise the enclosed area of land is it better to include the lake shore as part of the perimeter, or just ignore it and put the fence somewhere else? What shape fence do you choose to maximise the enclosed area of land? What is the maximal enclosed area of land possible? Last fiddled with by retina on 20210830 at 04:05 
20210830, 05:26  #10  
Romulan Interpreter
"name field"
Jun 2011
Thailand
2^{4}×613 Posts 
You will upset your neighbors, who have pythagorean triangle pieces of land ...
Quote:
Quote:
If no lake, the solution is obviously a circle/disk. Once the two disks (lake and perimetered land) touch, the area increase for a while as the centers of the two circles come closer, then decrease again. It obviously decrease after the point when the perimeter touches the lake in diametral points, but how it evolves in between, is unclear. A guess is that the maximum area will be when the perimeter touch the lake somewhere at 60 to 120 degrees angle at the center of the lake. Last fiddled with by LaurV on 20210830 at 05:35 

20210830, 09:24  #11 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2·23·137 Posts 

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