Optimizing Fencing for a One Square Mile Animal Pen

[Punkyc7]

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?

 

[tiny-tim]

Hi Punkyc7!

Hint: can you make the problem more symmetric?

 

[Punkyc7]

what is more symmetric then a semi circle

 

[tiny-tim]

exactly!

 

[Punkyc7]

Ok so it should be the semi circle.

 

[tiny-tim]

maybe and maybe not

you're thinking of making the solution more symmetric

i'm suggesting making the problem more symmetric ​

 

[Punkyc7]

hmm... I don't quite follow what you're hinting at.

 

[tiny-tim]

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?

 

[DryRun]

This is like one of those I.Q. test questions.

 

[Punkyc7]

divide by 2?

 

[DryRun]

divide by 2?

Divide what by 2?

 

[DryRun]

Since the OP has not responded, I'm going to venture a guess... The least amount of fencing needed is 3 miles?

 

[HallsofIvy]

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?

 

[Punkyc7]

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.

 

[DryRun]

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.

 

[DryRun]

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.

 

[Punkyc7]

so you think 3 is the correct answer?

 

[DryRun]

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.

 

[tiny-tim]

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 )

 

[DryRun]

OK, since the OP apparently gave up (and this question has been gnawing at me), I'm going to suggest my answer: $2\sqrt{\pi}-1$. Is it correct?

 

[tiny-tim]

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!

 

[DryRun]

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!

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?

 

[tiny-tim]

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

 

[DryRun]

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 $1/\sqrt{\pi}$.
Then, using the formula for finding the circumference of the circle: $2\pi (1/\sqrt{\pi})=2\sqrt{\pi}$. 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: $(2\sqrt{\pi}-1)$ miles.

 

[tiny-tim]

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

 

[DryRun]

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.