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#1 2012-06-22 23:20:57

amberzak
Member
Registered: 2012-03-16
Posts: 80

Minimising area of square and circle

A piece of string of length L is to be cut into two pieces. One piece is formed into a square, the other into a circle.
a)Where, if anywhere, would the string be cut to that the total area of the two shapes is a small as possible
Where, if anywhere, should the string be cut so the area is as large as possible

This is what I have:

Circle%20and%20Square.png

I'm assuming that the maximum would be just a circle, but I don't know how I'd show that. And not sure how I'd do the minimum


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#2 2012-06-22 23:25:03

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

Now find the derivative of that expression wrt X and set it equal to zero.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
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#3 2012-06-22 23:36:21

amberzak
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Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

How do I do that. Do I first need to get rid of the brackets? So it would be

x^2/16+Pi((L^2-x^2)/2pi^2)
?

Then where do I go? Can we take it a step at a time please?


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#4 2012-06-22 23:39:39

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

The expression you got now after pi is not correct.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#5 2012-06-22 23:43:31

amberzak
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Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

What did I do wrong?


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#6 2012-06-22 23:45:17

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

You squared the second bracket incorrectly.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#7 2012-06-22 23:46:52

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

Okay, what's the correct way to square it.


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#8 2012-06-22 23:48:47

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

Wait a sec till I turn on my computer.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#9 2012-06-22 23:54:29

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

Thanks anon.

Sorry, I've got the deadlines looming and I've burnt myself out


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#10 2012-06-23 00:02:09

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

Call me Stefy,please.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#11 2012-06-23 00:13:40

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

thanks. I obviously need to add x^2/16 to the end result you have there (I am guessing) and so do I do the cross multiplying?

Also, how do you so all that maths set up on the forum?


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#12 2012-06-23 00:17:51

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

Hi amberzak

Yes, just add x^2/16 to what I posted above. Then you will need to take the derivative of what you get.

If you want to know how I posted the expression in the above post, just go to LaTeX - A Crash Course. It is a sticky in the Help Me! section.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#13 2012-06-23 01:57:59

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

Right so I have got

So the df/dx is

Pray tell what do I do now?

(I know I have to equate it to mean 0, but what do I actually have to do?)

Last edited by amberzak (2012-06-23 02:03:00)


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#14 2012-06-23 02:08:06

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimising area of square and circle

Hi;

Equate it to 0 and you should have plotted the function.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#15 2012-06-23 02:11:09

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

Hey bobbym. Please see post above. How do I solve it equated to 0?


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#16 2012-06-23 02:15:35

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimising area of square and circle

Hi;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2012-06-23 02:25:34

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

Can you just explain what you did please?

I'm guessing that we divide through by x to get it on its own.


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#18 2012-06-23 02:33:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimising area of square and circle

Hi;

Times both sides by 8 pi.

Add 4L to both sides.

Factor the left.

Divided both sides by pi +4.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#19 2012-06-23 02:45:18

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

So then l-x=l-[4l/(pi+4)]
= L*Pi/(pi+4)

Is that the answer? That's the minimum right? How do I find the max? I imagine it is just using the whole length of rope for a circle, but how to I prove that? I can prove that a circle and a square of a set perimeter, the circle will have the biggest area.


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#20 2012-06-23 02:49:42

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimising area of square and circle

Hi amberzak;

I am sorry but I can not read that.

The x I gave is a minimum. I think the maximums will occur at an endpoint which will be
0 or L.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#21 2012-06-23 02:50:17

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Minimising area of square and circle

You should first check if that is the maximum or minimum by finding the second derivative and substituting x=4L/(pi+4)


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#22 2012-06-23 02:51:35

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

So your answer, bobby, that is the minimum? That is the answer?


Don't think outside the box. Think there is no box

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#23 2012-06-23 03:02:36

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimising area of square and circle

Hi;

Yes, that is my answer for a minimum.

The second derivative is,

which is positive.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#24 2012-06-23 03:04:02

amberzak
Member
Registered: 2012-03-16
Posts: 80

Re: Minimising area of square and circle

Thanks a million. big_smile


Don't think outside the box. Think there is no box

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#25 2012-06-23 03:06:04

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimising area of square and circle

Hi;

Please check post #23 and remember to thank anonimnystefy who did most of the work.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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