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#1 2013-03-25 22:46:18

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
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Minimise the distance

<< Problem attached >>

Now this is what I do:

Construct lines between  (1,5), (3,5), (4,9) and (1,9)
Now let us consider an arbitary point (1,y) and let the distance from (1,5) to (1,y) be r
Now we search for the value that would make ( 2^2 + r^2 ) + ((4 - r)^2 + 3^2) the minimum value
And the value of r which makes the above function to give its minimum is 2
So, is y = 5 + 2 the solution?

View Image: image.jpg

Last edited by Agnishom (2013-03-25 22:47:38)


'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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#2 2013-03-25 22:55:09

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

Hi;

I am getting 38 for a + b.


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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#3 2013-03-25 23:00:59

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
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Re: Minimise the distance

How to do it?


'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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#4 2013-03-25 23:10:59

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

For one thing always have a rough idea what the answer is. Sometimes you can have the exact answer before you even start with the the mathematics.


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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#5 2013-03-25 23:16:30

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

I believed it would be the midpoint


'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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#6 2013-03-25 23:21:19

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

A drawing also helps. Calculus will give an exact answer but geogebra got the answer to 15 places!

View Image: 2013-03-26_041334.gif

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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#7 2013-03-25 23:24:59

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

I used geogebra. But .... I couldn't find the exact

Okay, I'm ready for calculus


'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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#8 2013-03-25 23:29:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

y = 6.6 is the answer that geogebra comes up with.

This is the function to be minimized.

Can you do that?


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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#9 2013-03-26 00:00:20

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

So to minimise it we find its derivative f'
And then solve for f'(y) = 0
Because if that is 0 the slope is 0 and the horizontal line is formed and thats either the maxima or minima of the function

Am I correct?


'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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#10 2013-03-26 00:02:44

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

Yes, that is how calculus would do it. Again, I would urge you to graph the function first to have some idea where the minimum is.

My brother points out an nifty idea using vectors but I can not grasp it.


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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#11 2013-03-26 00:17:54

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

I haven't got an idea to find the derivative except using a CAS.
Actually, I am not fully acquinted with Calculus.

Is your brother on this forum?


'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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#12 2013-03-26 00:20:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

Yes, he is a member but he does not like forums.

If you can do it using a CAS that is okay too.


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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#13 2013-03-26 00:29:33

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

His username?


'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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#14 2013-03-26 00:42:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

jimmyR is his username.


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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#15 2013-03-26 01:02:40

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

Okay

The CAS didn't help me much. I haven't got a clue about equating that to 0 (by hand)

Last edited by Agnishom (2013-03-26 01:07:25)


'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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#16 2013-03-26 01:11:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

What CAS are you using?


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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#17 2013-03-26 01:15:19

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

MAXiMA,
I don't know much about it


'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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#18 2013-03-26 01:20:05

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,242

Re: Minimise the distance

The standalone or the one bundled with geogebra.


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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#19 2013-03-26 01:27:48

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,378

Re: Minimise the distance

OR

Reflect (3,5) in the line x=1 to give (-1,5)

Join this point to (4,9) to give the minimum distance.

EDIT: and the ratio is easily found by counting coordinate squares.

Bob

View Image: min dist.gif

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#20 2013-03-26 01:36:24

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

This sounds very interesting.
But:
1. That is not the answer
2. How do I know that that is the shortest distance?


'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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#21 2013-03-26 01:46:50

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,378

Re: Minimise the distance

hi Agnishom,

I have edited my hidden answer.  Hopefully, you'll like the correction better.

The distance (3,5) to (1,y) and the distance (-1,5) to 1,y) are the same by the properties of reflection.

So minimise by drawing a straight line from (-1,5) to (4,9)

I am finding the site very slow at the moment ... not sure if this is general or not.  It took 10 mins to make my last post, edit it and then correct the edit.

Hope what I have said is clear.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#22 2013-03-26 01:57:21

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

The site is okay for me

Yes, I got the idea Thanks a million smile


But I would be happier if you clarified it to be 33/5


'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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#23 2013-03-26 08:54:32

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,378

Re: Minimise the distance

Yes, sorry about that.  My excuse is that I was being harassed by the slow re-load time on pages.  It was driving me mad.  Originally I just posted the method.  Then I had the idea to work out the answer but hide it.  But I was completely off track because I had been thinking that 'y' divided the line in the ratio a:b ie 2:3 so I posted a + b = 5.

Then I read the question properly (why don't I do this first???) and revised my answer.  In my annoyance I thought 6 and 3/5 ... 6 5s are 35 add 3 makes 38.  So a/b = 38/5.  Big whoops.  I realised that must be wrong because that wasn't what my diagram was telling me, but it took ages to get the edit screen to work so I could do 6 5s are 30 add 3.    Pathetic excuse but there it is.  I've admitted to the forum that I cannot do my five times table.  I'll write it out 20 times.

Bob  smile

ps.  Load speed back to normal now.


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#24 2013-03-26 09:00:07

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,378

Re: Minimise the distance

FOR i = 1 TO 20
        j = 1
        REPEAT
          ans = j*5
          PRINT j;" x 5 = ";ans
          j = j + 1
        UNTIL j = 11
      NEXT i


Bob smile


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#25 2013-03-26 16:33:39

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,623
Website

Re: Minimise the distance

smile

I have problems with my multiplication tables too. Is it very important?


'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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