Optimizing Fencing for a One Square Mile Animal Pen

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A farmer can use one mile of a straight river as one border of an animal pen. What is the least amount of fencing needed to enclose a region of one square mile?

I'm guessing it is a semi circular pen. But I am not sure how to show that that is the minimum. I know its smaller than the triangle and the square.

Any advice on how to go about this?
 
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what is more symmetric then a semi circle
 
Ok so it should be the semi circle.
 
hmm... I don't quite follow what you're hinting at.
 
Punkyc7 said:
A farmer can use one mile of a straight river as one border of an animal pen. What is the least amount of fencing needed to enclose a region of one square mile?

What is asymmetric about the problem?

How would you get rid of the asymmetry? :wink:
 
This is like one of those I.Q. test questions. :smile:
 
Punkyc7 said:
divide by 2?

Divide what by 2?
 
Since the OP has not responded, I'm going to venture a guess... The least amount of fencing needed is 3 miles? :rolleyes:
 
That has nothing to do with the question the OP originally asked or was asked to respond to. What is more symmetric than a semi-circle? A circle.

In order that this be useful however do you know or have you proved that the figure containing a fixed area for the smallest perimeter (without any conditions) is a circle?
 
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I think that is a well know fact. But If I use a whole circle I am adding extra perimeter which is not what I want to do... I can't seem to figure out how I would use calculus to solve this.
 
HallsofIvy said:
In order that this be useful however do you know or have you proved that the figure containing a fixed area for the smallest perimeter (without any conditions) is a circle?

I've tried to prove this by finding the perimeter of a circle with radius, r, and fixed area (a constant value, say, 4 miles square). Then, tested the same fixed area with square, equilateral triangle and isosceles triangle. The results agree. I guess it's one of those secrets of geometry.
 
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Punkyc7 said:
I think that is a well know fact. But If I use a whole circle I am adding extra perimeter which is not what I want to do... I can't seem to figure out how I would use calculus to solve this.

The replies above seem to suggest finding the perimeter of the circle... and i think i solved the problem. :smile:
 
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so you think 3 is the correct answer?
 
Punkyc7 said:
so you think 3 is the correct answer?

No. I didn't take into account the fact that the circle has the least perimeter for a fixed area. I originally thought it was a square with one side = 1 mile, along the river. This is how i had (wrongly) calculated the required length of fencing to be the remaining 3 equal sides.
 
Punkyc7 said:
A farmer can use one mile of a straight river as one border of an animal pen. What is the least amount of fencing needed to enclose a region of one square mile?

The asymmetry is the river … the pen is only on one side of it.

Suppose the pen is allowed to be on both sides of the river, and to be two square miles …

how does the answer to that relate to the answer to the original question?​

(i assume this is the way Sharks answered it :wink:)
 
OK, since the OP apparently gave up (and this question has been gnawing at me), I'm going to suggest my answer: [itex]2\sqrt{\pi}-1[/itex]. Is it correct?
 
tiny-tim said:
it's a semi-circle …

any solution can be reflected in the river, and then has double the area and double the fence, with no river boundary! :wink:

You mean it's a similar semi-circle on each side of the river with diameter 1 mile?

What i thought was: From the area of 1 miles2, i get the radius r, which i then use to find the circumference of the circle. Then, minus 1 (length along river) from that circumference. It makes sense to me. What's wrong with that?
 
sharks said:
From the area of 1 miles2, i get the radius r, which i then use to find the circumference of the circle. Then, minus 1 (length along river) from that circumference.

sorry, i don't understand :confused:
 
My way of thinking does not involve crossing over to the other side of the river.
OK, we know the area of the circle is 1 miles2. From that area, i use the formula ∏r2 to get the radius of the circle, which is [itex]1/\sqrt{\pi}[/itex].
Then, using the formula for finding the circumference of the circle: [itex]2\pi (1/\sqrt{\pi})=2\sqrt{\pi}[/itex]. Now, if i consider a circle with that fixed area, and suppose its circumference is flexible, like a thin string border, and i align it with the river bank for a distance of 1 mile. The remaining length of the circle stays free, and although it won't be a circle anymore, it will still have a circular shape, which has length: [itex](2\sqrt{\pi}-1)[/itex] miles.
 
sharks said:
Now, if i consider a circle with that fixed area, and suppose its circumference is flexible, like a thin string border, and i align it with the river bank for a distance of 1 mile. The remaining length of the circle stays free, and although it won't be a circle anymore, it will still have a circular shape …

i don't understand :confused:
 
I realize now that my method is flawed. Since the shape of the circle changes, despite keeping a fixed circumference, the area also decreases. So, your method is indeed the only correct one. :redface:
 

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