Math Is Fun Forum

nha

Member

Registered: 2010-09-05

Posts: 43

Optimal Design question

Hello math lovers, can someone please provide me some help with an optimal design question. Any sort of help would be awesome.

Q1) Show that among all the 4-sided polygons inscribed in a circle the squares have the largest area.
Q2) And for fixed n which is the n-gon of largest area inscribed in a circle?

For Q1 I know I should focus on showing that all sides of a quadrilateral of maximal area have to be of equal length. So I would use Brahmagupta's formula I think, but how would I show that the squares have the largest area?

Q2) I don't know how to approach this.

Thank you all for any help.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Optimal Design question

Hi nha;

Welcome to the forum. Start by making a good diagram . Draw a circle and put a rectangle inside of it.

You want to get everything in terms of one variable x:

Area as you know is X * Y, so:

Now you differentiate with respect to x. I leave the steps of the
differentiation up to you to do. I just provide the answer. Please work it out yourself.

Set the derivative equal to 0.

Solve for x, I leave that to you also. I have given you the answer. Try to solve the equation by yourself.

Reject the negative answer. Plug into equation 1) to get y.

Since x = y ( look at the drawing we have a square. ) Now you must prove that the critical point is a maximum and you are done. Please try this yourself.

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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.

nha

Member

Registered: 2010-09-05

Posts: 43

Re: Optimal Design question

Thanks so much bobbym. It seems so easy now, thanks for the huge help. I proved it was a maximum.

Any thoughts/help on Q2? I believe it should be a regular n-gon but I don't know how to go about proving this.

Last edited by nha (2010-09-05 17:57:01)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Optimal Design question

Hi nha;

Your welcome, for Q2, I am not sure that I understand the question.

---

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.

nha

Member

Registered: 2010-09-05

Posts: 43

Re: Optimal Design question

Hi nha;

Your welcome, for Q2, I am not sure that I understand the question.

I think it is asking to find the n-gon that has the largest area inscribed in the circle, but it would have to be an answer like 'the regular n-gon' otherwise it doesn't make sense to me. Maybe because as n gets larger, a regualr n-gon approximates to a circle??

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Optimal Design question

Hi nha;

Yes, I agree, the answer is a regular polygon.

Maybe because as n gets larger, a regualr n-gon approximates to a circle??

Yes, the old Archimedes idea to estimate pi. Only thing is I am not coming up with anyway to prove that each side abd each angle are equal.

Here is a an article in pdf form that covers your question.

http://www2.edc.org/mathproblems/proble … xArea3.pdf

---

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.