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Old 2021-08-29, 06:02   #1
MattcAnderson
 
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"Matthew Anderson"
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Smile 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
File Type: pdf farmer Fred_s field exercise.pdf (152.9 KB, 30 views)
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Old 2021-08-29, 06:15   #2
retina
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Quote:
Originally Posted by MattcAnderson View Post
... 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?
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Old 2021-08-29, 06:56   #3
a1call
 
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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
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Old 2021-08-29, 07:34   #4
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Quote:
Originally Posted by retina View Post
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
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Old 2021-08-30, 01:39   #5
MattcAnderson
 
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Hi again all,

Hopefully this classifies this exercise. I want to make it clear to everybody.
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Old 2021-08-30, 02:23   #6
kriesel
 
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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
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Old 2021-08-30, 02:48   #7
retina
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Quote:
Originally Posted by kriesel View Post
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.
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Old 2021-08-30, 03:51   #8
LaurV
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Quote:
Originally Posted by MattcAnderson View Post
There are right angles at each corner of this field.
Quote:
Originally Posted by retina View Post
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
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Old 2021-08-30, 04:04   #9
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Quote:
Originally Posted by LaurV View Post
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
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Old 2021-08-30, 05:26   #10
LaurV
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You will upset your neighbors, who have pythagorean triangle pieces of land ...
Quote:
Originally Posted by retina View Post
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
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Old 2021-08-30, 09:24   #11
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Quote:
Originally Posted by LaurV View Post
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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