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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Four circles of radius 1 cm are drawn with their centres at the four verticies of a square with side length 1 cm. Find the area, in square centimetres, of the region overlapped by all four circles.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

Hi Jane;

*Last edited by bobbym (2009-10-20 03:09:04)*

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**TheDude****Member**- Registered: 2007-10-23
- Posts: 361

Edit: I get the same answer as bobby except for his 1/3 multiplier. I can't see where that's coming from.

*Last edited by TheDude (2009-10-20 02:10:33)*

Wrap it in bacon

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

TheDudes answer is what I got.

Now can you guys work out the problem without using integral calculus?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

Hi;

Had to adjust this post because apparently I can't copy and paste correctly.

These agree with this answer although they do it in a better way.

http://faculty.missouristate.edu/l/lesreid/Adv08.html

http://www.askmehelpdesk.com/mathematic … 108-3.html

*Last edited by bobbym (2009-10-20 03:21:34)*

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

bobbym wrote:

If you didn't get

I got

As did Sophie: http://www.thescienceforum.com/viewtopi … 726#214726

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**TheDude****Member**- Registered: 2007-10-23
- Posts: 361

I feel sorry for people who had to get through life before calculus.

*Last edited by TheDude (2009-10-20 03:06:16)*

Wrap it in bacon

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

Hi Jane and Dude;

Sorry!!!!! I have a typo! The 1/3 does not belong there. Did cancelling inside the parentheses and left the 1/3. Fixing the other post. Jeez, got the right answer and mucked it up copying it over.

Anyway the calculus way was easier than the Sophie way.

*Last edited by bobbym (2009-10-20 03:27:10)*

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

TheDude wrote:

I feel sorry for people who had to get through life before calculus.

*Last edited by JaneFairfax (2009-10-20 22:39:39)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

Hi Jane;

Jane wrote:

In any case, I do not find the calculus approach easier than Sophies solution.

You're not saying that her method is easier so can I presume they are of equal difficulty?

Jane wrote:

Some people clearly need to have their heads checked!

Comments like that hurt my feelings.

What do you have against calculus and why do you find her method easier?

*Last edited by bobbym (2009-10-20 18:12:25)*

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Fun fact: If two people have different opinions, it's not necessarily true that one of them is crazy.

Insults aside, it's a good discussion though. I suppose it depends what you mean by easier.

Lots of geometry problems like this one can be solved with calculus, so for some people, the first reaction on seeing this problem would be to set up an integral. In this case, it's easy to see *how* to do the problem, but doing the calculation might be trickier.

Other people, like Sophie, try to come up with a geometric way. The way she splits the 'complement region' into 4 congruent shapes is a beautiful idea, and I really like her method. However, she could only come up with that method after seeing this particular problem, and a similar question might not be solvable in the same way.

For people who can see geometric solutions like that easily though, I agree that getting to avoid the nasty integrals is a nice bonus.

Why did the vector cross the road?

It wanted to be normal.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

bobbym wrote:

Jane wrote:Some people clearly need to have their heads checked!

Comments like that hurt my feelings.

I wasnt referring to you; I was referring to The moronic Dude retard before you.

mathsyperson wrote:

Fun fact: If two people have different opinions, it's not necessarily true that one of them is crazy.

Insults aside, it's a good discussion though. I suppose it depends what you mean by easier.

Lots of geometry problems like this one can be solved with calculus, so for some people, the first reaction on seeing this problem would be to set up an integral. In this case, it's easy to see

howto do the problem, but doing the calculation might be trickier.Other people, like Sophie, try to come up with a geometric way. The way she splits the 'complement region' into 4 congruent shapes is a beautiful idea, and I really like her method. However, she could only come up with that method after seeing this particular problem, and a similar question might not be solvable in the same way.

For people who can see geometric solutions like that easily though, I agree that getting to avoid the nasty integrals is a nice bonus.

Totally agree!

*Last edited by JaneFairfax (2009-10-21 20:43:17)*

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**TheDude****Member**- Registered: 2007-10-23
- Posts: 361

mathsyperson wrote:

Lots of geometry problems like this one can be solved with calculus, so for some people, the first reaction on seeing this problem would be to set up an integral. In this case, it's easy to see

howto do the problem, but doing the calculation might be trickier.

That's a really good point. While I doubt I could have ever come up with the geometric method for solving the problem, I would have been in similar trouble if I didn't have easy access to a list of common integrals and had to integrate

myself.Wrap it in bacon

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

Hi Jane;

Comments like that hurt my feelings.

Was just probing to see who you were talking about, "some people", might not have meant me. Glad to see that you did mean me. I am not interested in conforming to the world's view of normalcy.

Though you should probably be well enough acquainted with me now to take my occasional outbursts seriously,

You, left out the not so... Again, no problem. I think we understand each other and I am comfortable with that, so no apology is necessary. Continue to share your viewpoints.

*Last edited by bobbym (2009-10-21 03:22:48)*

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

Hi All;

mathsyperson wrote:

Fun fact: If two people have different opinions, it's not necessarily true that one of them is crazy.

Patently incorrect. Should read, whenever someone has a different opinion, quietly suspect that they are insane. Do not share this assessment with them, lest you enrage that crazy person. - Old proverb.

The following is not meant to insult anyone's methods. I believe in doing what works for you.

Villenkin speaks of the "Teakettle principle." Loosely it goes like this:

A physicist and a mathematician are given an empty teakettle, a fire and a water supply and are asked to boil water. They both fill the teakettle, place it on the fire and boil the water. Now they are both given a teakettle already filled with water. The physicist after much thought shouts "Eureka" and places the teakettle on top of the fire and boils the water. The mathematician immediately empties his teakettle, the physicist asks why did you empty it? The mathematician says, "because I already know how to solve the empty teakettle problem."

It is nice to think up a beautiful and novel solution for every problem but a general method is more useful. Reducing a problem to one you already know how to solve like the integration of a function. Sophie's method parallels the physicists way in the story. A new brainstorm is required for each problem. The integration is a general idea, applicable to many problems of this type.

TheDude wrote:

I would have been in similar trouble if I didn't have easy access to a list of common integrals and had to integrate ...

Those integrations if you could not have done them analytically, would have responded very well to numerical integration, to any number of digits that you require.

*Last edited by bobbym (2009-10-21 04:50:08)*

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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What do you have against calculus and why do you find her method easier?

Non-Calculus is legion. It is something similar to the argument of using computers

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 85,215

There are several books out there for doing calculus problems without using calculus.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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