Calculus Mazimizing Problem

1. May 5, 2012

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 im not sure how to show that that is the minimum. I know its smaller than the triangle and the square.

2. May 5, 2012

tiny-tim

Hi Punkyc7!

Hint: can you make the problem more symmetric?

3. May 5, 2012

Punkyc7

what is more symmetric then a semi circle

4. May 5, 2012

exactly!

5. May 5, 2012

Punkyc7

Ok so it should be the semi circle.

6. May 5, 2012

tiny-tim

maybe and maybe not

you're thinking of making the solution more symmetric

i'm suggesting making the problem more symmetric

7. May 5, 2012

Punkyc7

hmm... I dont quite follow what you're hinting at.

8. May 5, 2012

tiny-tim

What is asymmetric about the problem?

How would you get rid of the asymmetry?

9. May 5, 2012

sharks

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

10. May 5, 2012

Punkyc7

divide by 2?

11. May 6, 2012

sharks

Divide what by 2?

12. May 6, 2012

sharks

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

13. May 6, 2012

HallsofIvy

Staff Emeritus
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?

Last edited: May 6, 2012
14. May 6, 2012

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 cant seem to figure out how I would use calculus to solve this.

15. May 6, 2012

sharks

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.

Last edited: May 6, 2012
16. May 6, 2012

sharks

The replies above seem to suggest finding the perimeter of the circle... and i think i solved the problem.

Last edited: May 6, 2012
17. May 6, 2012

Punkyc7

so you think 3 is the correct answer?

18. May 6, 2012

sharks

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.

19. May 6, 2012

tiny-tim

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 )

20. May 9, 2012

sharks

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?