Minimising area of square and circle

amberzak · 2012-06-22 23:20:57

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

anonimnystefy · 2012-06-22 23:25:03

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

amberzak · 2012-06-22 23:36:21

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?

anonimnystefy · 2012-06-22 23:39:39

The expression you got now after pi is not correct.

amberzak · 2012-06-22 23:43:31

What did I do wrong?

anonimnystefy · 2012-06-22 23:45:17

You squared the second bracket incorrectly.

amberzak · 2012-06-22 23:46:52

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

anonimnystefy · 2012-06-22 23:48:47

Wait a sec till I turn on my computer.

amberzak · 2012-06-22 23:54:29

Thanks anon.

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

anonimnystefy · 2012-06-23 00:02:09

Call me Stefy,please.

amberzak · 2012-06-23 00:13:40

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?

anonimnystefy · 2012-06-23 00:17:51

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.

amberzak · 2012-06-23 01:57:59

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)

bobbym · 2012-06-23 02:08:06

Hi;

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

amberzak · 2012-06-23 02:11:09

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

bobbym · 2012-06-23 02:15:35

Hi;

amberzak · 2012-06-23 02:25:34

Can you just explain what you did please?

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

bobbym · 2012-06-23 02:33:27

Hi;

Times both sides by 8 pi.

Add 4L to both sides.

Factor the left.

Divided both sides by pi +4.

amberzak · 2012-06-23 02:45:18

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.

bobbym · 2012-06-23 02:49:42

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.

anonimnystefy · 2012-06-23 02:50:17

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

amberzak · 2012-06-23 02:51:35

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

bobbym · 2012-06-23 03:02:36

Hi;

Yes, that is my answer for a minimum.

The second derivative is,

which is positive.

amberzak · 2012-06-23 03:04:02

Thanks a million.

bobbym · 2012-06-23 03:06:04

Hi;

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

Math Is Fun Forum

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

Minimising area of square and circle

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

Re: Minimising area of square and circle

#3 2012-06-22 23:36:21

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

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

Re: Minimising area of square and circle

Board footer