# 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

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 by a moderator: 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?