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## #1 2006-04-17 01:13:49

Joey
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### Stuck on cone max-volume (picture)

A cone is going to be build with 6m long sticks as the picture shows, what is the maximum volume of the cone? http://hem.bredband.net/nimkam/cone.jpg
http://hem.bredband.net/nimkam/cone.jpg
The picture shows the formula for the volume of a cone, it is obvious that the radius and height varies when the 2 6m long sticks move but I have no clue on how to proceed. I need a formula so I can derive it V to get the maximum volume but pointers on how to get that formula are welcomed.  picture link

## #2 2006-04-17 02:43:07

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Stuck on cone max-volume (picture)

Just to make sure I understand the problem...

You are taking 6m long sticks, and leaning them in towards each other to make the cone.  But you can vary the angle at which the lean in.  Thus, the height and base radius are variables and you wish to see which height and radius will make the largest volume cone.

Right?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #3 2006-04-17 02:52:18

Ricky
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Registered: 2005-12-04
Posts: 3,791

### Re: Stuck on cone max-volume (picture)

I'm going to assume that you are in Calculus, as that seems the easiest way to do the problem.

The major problem in this problem, is the problem of having a function of two variables.  Finding the min/max of such a function is fairly difficult, especially when compared to finding the min/max of a function of one variable.  So we want to convert this problem into that of one variable.

If you think about it, it already is.  Like I said in the above post, you can only control one thing, the angle at which the stick are at.  This, in turn, controls the radius and the height of the cone.  So to make it a function of one variable, lets calculate the radius and height by the angle alone:

This just uses simple trig.  Ah, now all we have to do is substitute it in:

And now we have a function of one variable.  Take the derivative, find the zeros, and then find the max.

Last edited by Ricky (2006-04-17 02:53:25)

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #4 2006-04-17 02:58:43

Joey
Guest

### Re: Stuck on cone max-volume (picture)

Thank you! Yes I´m in calculus, yet we havent studies cos , tan and sin yet so I assumed thered be a solution without involving those in this problem. But yes, you assume right, those two sticks are to build the cone. Thank you yet again

## #5 2006-04-17 03:23:30

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,201

### Re: Stuck on cone max-volume (picture)

Without trignometry? Let me try...
h²+r²=6²
(from the picture)

Therefore,
h²=36-r²

Substituting the value of h from the above equation,

Now there is only one variable,
use the condition for maximum value, and the problem is solved!

Character is who you are when no one is looking.

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## #6 2006-04-17 04:02:45

Joey
Guest

### Re: Stuck on cone max-volume (picture)

Thank you Ricky and Ganesh, I looked the basics of sin, tan and cos up after Ricky's through answer and learned a few things but still haven't learned to derive those 3 so I´ll have to go with Ganes method. Amazing that it can be solved in at least 2 presented ways. Thank you yet again. Keep up the good work, your help is invaluable to many!

Best regards Joey

## #7 2006-04-17 06:52:28

Ramyar
Member
Registered: 2006-03-12
Posts: 6

### Re: Stuck on cone max-volume (picture)

Go on www.cartoonnetwork.com then go on courage the cowardly dog then play the game

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## #8 2006-04-17 06:59:13

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

### Re: Stuck on cone max-volume (picture)

Ramyar wrote:

Go on www.cartoonnetwork.com then go on courage the cowardly dog then play the game

am I the only one who doesnt get this? Is it a joke, or?

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## #9 2006-04-17 07:04:38

Joey
Guest

### Re: Stuck on cone max-volume (picture)

No Patrick, were in the same boat on that one. :S

## #10 2006-04-17 15:17:15

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,582

### Re: Stuck on cone max-volume (picture)

The angle is not 45 degrees because a 3,4,5 triangle cone has a larger volume.
It seems as though perhaps there are two answers, maybe one less than 45 degrees at the vertex (top of cone), and
another answer greater than 45 degrees, however, this is only a conjecture.
Certainly, the larger base ( > 45 degrees) is an answer, but I still wonder if a tall skinnier than 45 could also get bigger in volume before getting smaller again.  This would have to be looked into in more depth than I have been thinking simply abstractly.
The angle found may be the same as the angle on a side of a cubic rectangle with max volume as well, but not sure yet...

Last edited by John E. Franklin (2006-04-17 15:17:46)

igloo myrtilles fourmis

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## #11 2006-04-17 16:45:59

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Stuck on cone max-volume (picture)

John, you get the maximum of the function by taking the derivative and setting it to 0.  You end up with 4.8989 if you use ganesh's, and 0.6154 if you use mine.  But remember, 6cos(0.6154) = 4.8989.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #12 2006-04-17 17:07:32

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

### Re: Stuck on cone max-volume (picture)

A theorem:

(
is true when a=b=c)

One Application:

(
is true when r²/2=r²/2=36-r² or r=2√6)

volumn reaches its maximum while f(r) reaches its own.

Last edited by George,Y (2006-04-17 17:23:00)

X'(y-Xβ)=0

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## #13 2006-04-17 17:28:17

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

### Re: Stuck on cone max-volume (picture)

2√6≈4.8989
the theorem is famous in the field of inequalities, which is neglected by standard math education.

X'(y-Xβ)=0

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## #14 2006-04-17 17:42:45

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

### Re: Stuck on cone max-volume (picture)

John, good try. you may plot ganesh's complicated function using a grapher.

X'(y-Xβ)=0

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## #15 2006-04-17 17:50:34

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

### Re: Stuck on cone max-volume (picture)

you can plot "y=x*x*sqrt(36-x*x)" for xMin=0 to xMax=6 and parameters none at this site:

http://www.wessa.net/math.wasp

Last edited by George,Y (2006-04-17 17:52:38)

X'(y-Xβ)=0

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## #16 2006-04-27 02:15:41

Mozartmoses
Member
Registered: 2006-04-26
Posts: 10

### Re: Stuck on cone max-volume (picture)

Is the number [(44+\sqrt{1996})^{100}] odd or even?

I am confused while looking at it, help me to approach the problem to get the answer.

With regards
Moses

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## #17 2006-04-27 02:24:23

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,201

### Re: Stuck on cone max-volume (picture)

Mozartmoses,

I shall rewrite the expression from what I understand.

Since 1996 is not a perfect square, the value of square root of 1996 is an irrational number.
Hence, the resultant is neither even nor odd, it is an irrational number.

The expansion of

would contain 101 terms, many of the terms would be irrational numbers.
For example, the second term of the expansion would be

which is an irrational number!

Character is who you are when no one is looking.

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## #18 2006-04-27 02:29:41

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Stuck on cone max-volume (picture)

The root of 1996 is irrational, and the entire expression will be as well.  Even and odd don't make any sense when you are talking about something with a decimal place.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #19 2006-05-02 00:33:27

Mozartmoses
Member
Registered: 2006-04-26
Posts: 10

### Re: Stuck on cone max-volume (picture)

Hey !!

I got it now. Thanks for your help.

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## #20 2006-05-02 01:14:22

Mozartmoses
Member
Registered: 2006-04-26
Posts: 10

### Re: Stuck on cone max-volume (picture)

Victoria woke up in the middle of the night and looked at her digital clock, it said 2:58, then she saw it change to 2:59 and then 3:00. Bored she began adding up the digit as they changed 15,16.3. Hmmmm
At 3:01 the digit added up to 4. A minute later the sum was 5. Victoria liked the pattern. She turned on the light and began figuring. She found that of all the digit sums on a 12 hour digital clock only one is unique meaning there is only one time that give you that sum.

What is the unique digit sum, and what is it's corresponding time?

I know, you people will come up with a solution.

With regards
Moses

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## #21 2006-05-02 01:50:34

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,201

### Re: Stuck on cone max-volume (picture)

Moses,
09:59 would give a sum of 23, a unique sum, I guess!

Character is who you are when no one is looking.

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## #22 2006-05-02 01:55:23

Mozartmoses
Member
Registered: 2006-04-26
Posts: 10

### Re: Stuck on cone max-volume (picture)

Hi !!

I too guessed the same, since i am not sure of the answer, i asked you.
Thanks a lot Ganesh.

Good work

-------  Moses

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## #23 2006-05-02 09:26:37

MathsIsFun
Registered: 2005-01-21
Posts: 7,555

### Re: Stuck on cone max-volume (picture)

And on a 24 hour clock you get two uniques: 0:00 gives 0, and 19:59 gives 24

Isn't it weird that a 24-hour clock's maximum digit sum is also 24?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #24 2006-05-03 07:01:21

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

### Re: Stuck on cone max-volume (picture)

The 12-hour clock has 2 solutions as well. 1:00 is the only time that gives a sum of 1.

Why did the vector cross the road?
It wanted to be normal.

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## #25 2006-05-08 05:34:59

Mozartmoses
Member
Registered: 2006-04-26
Posts: 10

### Re: Stuck on cone max-volume (picture)

What is the difference between sequence and pattern? Can any one help me to get a clear picture?