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2021-08-29, 06:02   #1
MattcAnderson

"Matthew Anderson"
Dec 2010
Oregon, USA

3C716 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
Attached Files
 farmer Fred_s field exercise.pdf (152.9 KB, 43 views)

2021-08-29, 06:15   #2
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

2×23×137 Posts

Quote:
 Originally Posted by MattcAnderson ... the field is in the shape of a rectangle. ... the 3 sides of his field ...
One of these things is not like the others.

a rectangle is a quadrilateral with four right angles

Do you mean a triangle?

 2021-08-29, 06:56 #3 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 1000011111102 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 2021-08-29 at 06:58
2021-08-29, 07:34   #4
axn

Jun 2003

121158 Posts

Quote:
 Originally Posted by retina One of these things is not like the others. a rectangle is a quadrilateral with four right angles Do you mean a triangle?
Three sides need fencing. The fourth side is a river, which doesn't need fence.

EDIT:-

Let a,b,a be the three sides
2a+b=200
Maximize Area = ab = a*(200-2a) = 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 2021-08-29 at 07:40

 2021-08-30, 01:39 #5 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 967 Posts Hi again all, Hopefully this classifies this exercise. I want to make it clear to everybody. Attached Thumbnails
 2021-08-30, 02:23 #6 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 172516 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 2021-08-30 at 03:00
2021-08-30, 02:48   #7
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

11000100111102 Posts

Quote:
 Originally Posted by kriesel Most livestock will not respect a waterway as a fence.
Fences can be used for many purposes. Livestock containment is just one of them.

It could be a decorative picket fence for Fred to build a house inside.

2021-08-30, 03:51   #8
LaurV
Romulan Interpreter

"name field"
Jun 2011
Thailand

24·613 Posts

Quote:
 Originally Posted by MattcAnderson There are right angles at each corner of this field.
Quote:
 Originally Posted by retina Do you mean a triangle?

Bwaaa haha!

(sorry, I could not stop it!)

Last fiddled with by LaurV on 2021-08-30 at 03:52

2021-08-30, 04:04   #9
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

630210 Posts

Quote:
 Originally Posted by LaurV Bwaaa haha!
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 2021-08-30 at 04:05

2021-08-30, 05:26   #10
LaurV
Romulan Interpreter

"name field"
Jun 2011
Thailand

24·613 Posts

You will upset your neighbors, who have pythagorean triangle pieces of land ...
Quote:
 Originally Posted by retina There are right angles, there are wrong angles, and there are triangles.
You forgot left angles.

Quote:
 Originally Posted by retina 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,
Yes it is.

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 2021-08-30 at 05:35

2021-08-30, 09:24   #11
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

2×23×137 Posts

Quote:
 Originally Posted by LaurV Once the two disks (lake and perimetered land) touch, the area increase for a while ...
Are you sure that some other shape, that is not a simple circle segment, won't give more area?

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