Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Bitter areas.

Hi;

3 listening posts are situated in the north atlantic. They form an equilateral triangle. A nuclear explosion takes place 500 miles from one listening post, 1200 miles from another and 1300 miles from the last listening post. The explosion takes place inside the triangle that the posts form. What is the area of the triangle that the listening posts form?

A says) Impossible to determine, Cauchy wrote a paper on it and I am well familiar with that paper. It is a great advantage that I enjoy knowing more theory than B does. I know when a problem can not be solved, B does not. That is why I can work on the ones that can be solved.

B says) Hold on A, the problem is easy using...

C says) Cauchy hmmm, I have heard of him.

D says) Yes, he owned a bakery on my block and made good cannoli.

C says) Weren't they first discovered by Schiaperelli?

A says) Yes, I read a paper about the cannoli's on Mars, they were found to be a hoax. Something B probably knows all about and believes.

E says) What the heck are you talking about? I hate idiots and A,C and D qualify. I have already solved the problem.

Can you?

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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.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,913

Re: Bitter areas.

Hi Bobby,

Is it...

Last edited by phrontister (2015-02-07 00:58:08)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

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.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,913

Re: Bitter areas.

Hi;

Last edited by phrontister (2016-10-09 19:42:45)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

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.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,913

Re: Bitter areas.

Hi;

Last edited by phrontister (2015-02-08 03:10:34)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

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.

Matir

Member

Registered: 2016-10-08

Posts: 22

Re: Bitter areas.

Hey, isn't the Earth round?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Not really, the spin distorts the shape. For the sake of the problem a flat plane was considered okay.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Bitter areas.

Well. I am getting analytically the same result as phrontister. There is also a solution for the blast to occur outside the triangle formed by the posts.Then the area will be

sq.miles.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

If you got the same answer as phrontister you have solved the problem. Good work. The other answer violates the terms of the problem.

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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.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Bitter areas.

Hello:

Quite elementary.

Last edited by ElainaVW (2016-10-09 21:10:39)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

Yes, that is a solution and it makes good use of the new Region commands. Very good.

---

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.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Bitter areas.

How do you say it, yall gots to come up with something a bit harder aye bumpkin.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Bitter areas.

Hello:

Quite elementary.

Is it elementary or elementri containing 3 Elements?

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Bitter areas.

Hello:

How about too easy. That will take care of tary-try-tree-tri-trey.

mycalsuite

Member

Registered: 2025-09-10

Posts: 79

Website

Re: Bitter areas.

A is right; there isn't enough data to identify the area in a unique way.

The equilateral triangle's size is not uniquely fixed by the distances to its three vertices.
As a result, the area cannot be calculated using the information provided.

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mycalsuite

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,913

Re: Bitter areas.

Hi mycalsuite;

A is right; there isn't enough data to identify the area in a unique way.

The equilateral triangle's size is not uniquely fixed by the distances to its three vertices.
As a result, the area cannot be calculated using the information provided.

By "not uniquely fixed", do you mean that multiple solutions are possible?

If so, do you have an example you could post?

Bobbym (OP), thickhead, ElainaVW and I have all arrived at the same single solution (ElainaVW's 2500√3(169+60√3) being slightly less simplified than the others' 2500(180+169√3).

Here's the Geogebra drawing I mentioned in post #4:

The Geogebra measurements check out...

EDIT: I googled this puzzle yesterday, and found that it has appeared in different guises over many years (more recently on websites such as Artofproblemsolving, math.Stackexchange and Reddit). The three internal-point-to-vertex lengths are often a Pythagorean triple (bobbym's {500, 1200, 1300} is one), opening up the use of a particular solution strategy. All solutions I found are single.

Last edited by phrontister (Yesterday 19:41:14)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,913

Re: Bitter areas.

The three internal-point-to-vertex lengths are often a Pythagorean triple (bobbym's {500, 1200, 1300} is one), opening up the use of a particular solution strategy.

One such strategy yields the solution without using CAS and without knowing (a) the location of the explosion and (b) the side lengths of the equilateral triangle formed by the listening posts.

Here's the idea:

My previous post shows a large equilateral triangle ABC comprising 3 smaller inner triangles (blue, green, gold).

Place one copy of each of these 3 small triangles outside triangle ABC, arranged as follows:
   - The longest side of the outer triangles should lie exactly along a side of ABC, with each on its own ABC side (Geogebra has a handy tool for that).
      Note: This results in a hexagon consisting of triangle ABC and its 3 attached outer triangles.

Draw 3 strategically placed lines inside the hexagon, so that the hexagon comprises 3 equilateral triangles and 3 congruent right-angled triangles. The 3 newly added lines have a critical role, and must be proven valid.

Calculate the areas of the 3 equilateral triangles and 3 right-angled triangles, and sum the results to obtain the area of the hexagon.

The area of triangle ABC is half the hexagon's area, and is 2500(180+169√3)...as per earlier posts.

Last edited by phrontister (Yesterday 19:41:59)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson