Water Access.

[onta...@hotmail.com]

A submersible tank is stationary in deep water. It is shaped like two opposing right cones with a cylindrical section between them.
The cones are 6ft. diameter by 4ft.in length, and the cylindrical section between the cones is 6ft. in diameter by 4ft. in length and thus making the tank's overall length 12ft.
A diver is attached to one of the cones apex by a 14ft. lifeline, which just allows him to reach the other cone apex.
What volume of water does the diver have access to?.

[Henry]

It is unclear but the "14 ft. lifeline, which just..." suggests the
diver is outside the tank, so the answer is the volume of the sphere,
less the volume of the tank.

[onta...@hotmail.com]

No,the outline of the tank prevents the formation of a true sphere by the lifeline, although the 3D form generated by the lifeline can still can encompassing the tank.

[João Pedro Afonso]

My candidate solution: no less than 10986.987 m^3, unless the diver is very very tiny, in which case it might be rounded to that number :-)

I'm surprised no one tried to present a solution until now. It is not that difficult, I think. Thanks mates (Philippe92, Avni, etc) for pointing me out and explain the Pappus theorem some years ago.

Cheers,
JPA

[onta...@hotmail.com]

Hello again Joao,
The access volume is obviously < [4/3*pi*14^3 - 5/3*pi*4*3^2]

Let t = atan(.75)
Area (1) = (180 - t)/360*pi*14^2
Area (2) = (t)/360*pi*9^2
Area (3 = (t)/360*pi*5^2

The position of each centroid along the interior angle bisectors of each of these three areas is given as 38.197*r*sin(x)/x, where x is half the sector angle. It is necessary to convert these to their transverse values (perpendicular to the tank axis) to find the diameters of rotation.
Regards
Bill

[João Pedro Afonso]

> Hello again Joao,
> The access volume is obviously < [4/3*pi*14^3 -
> 5/3*pi*4*3^2]

My previous result passed that test... but I made a childish mistake in the area calculi (used angle*R^2 instead of angle*R^2/2... I was unsure for a time what angle I was going to use and probably mixed my choices). If I didn't do more errors, the correct result should be 11240.5.

[onta...@hotmail.com]

I made the result 11321 after a lot of messy calculations.

[João Pedro Afonso]

> I made the result 11321 after a lot of messy
> calculations.

But Bill, your result doesn't conform with your own validation criteria (be less than 11305.5) :-)

[onta...@hotmail.com]

Joao
Yes, I made a mistake on the smallest of the three volumes. Revising and rounding off I now get 10972 c.m.

[João Pedro Afonso]

What we need here is a third opinion, but being the problem one month old, and having you (and me) tip it off heavily, I'm not sure if we are going to get any. I'll try to break my solution in parts to see where we diverge (I'll use also your past answer as a template). First some formulas to use:

Pappus theorem

V = 2PiR*A

The figure to which I'll apply the formula will be always the same, a circular sector with angle between sides of

> Let t = atan(.75)

This figure has an area equal to tR^2/2=0.32175R^2 (t in radians). It's centroid is 4R(Sin(t/2))^2/(3t)=0.2072R away from its sides

> Area (1) = (180 - t)/360*pi*14^2

Total volume:
V = 4PiR^3/3 - 2Pi 0.2072R *0.32175R^2 with R=14
= 10344.6

> Area (2) = (t)/360*pi*9^2

V = 2Pi (0.2072R+3) *0.32175R^2 with R=9
= 796.6

> Area (3 = (t)/360*pi*5^2

V = 2Pi (3-0.2072R) *0.32175R^2 with R=5
= 99.3

10344.6
796.6
+99.3
----------
11240.5

Which is the number I presented to you. My reasoning should be obvious so, if I failed, it should be easy to point why.

Cheers,
Joao Pedro Afonso

[João Pedro Afonso]

> Joao,
> No need for another opinion, you are correct !!
> Although I got the correct method, I made several
> wrong calculations.
> Cheers.

A third opinion here means another one tried to do the problem. :-)

Wrong calculations appears to be the pettiest (and forgettable) motive to fail a problem, after all, the hard part is deducing the solution, isn't? Unfortunately, pettiest or not, is my most common cause to fail them too... which is embarrassing because here, I present always the result first (as everyone, to not stole others the pleasure of solving the problems by their one), and with only it, is impossible to anyone knew where I missed. So I tried to do the calculations several times, to see if they agree. The best method is still to lay down all the expressions in some programmable machine or software, and do everything in one step, but I forget to do that sometimes (call it the paper and pencil itching). This problem was no exception and between the several results I got until they stabilized, it was with some fear I presented one.

Now, comes the funny part. Your answer about my first candidate "guess" was enough cryptic to me (you didn't said I was wrong but neither that I was right) to lead me to think there was something amiss in my solution. As it was, there was: an angle which meaning I changed at mid-reasoning, which changes I didn't properly transport to all the expressions. I thought you were trying to seduce answers from others too before judging it, but in a chance you were saying I was wrong in a nice way, I checked my work. To me, is funny to think I found my error based on a comment you would produced even if I was right... because it was also based in faulty calculations :-) ...call it serendipity, I suppose.

I'm sorry for this long comment, Bill. I've been missing from this newsgroup for a while, and sometimes I need to write some lengthy piece of philosophy about ways of reasoning, to check mines. Two last thoughts: numbers maybe wrong while the deductions are right and the reverse may happen too. Numbers are also boring while deductions are not. Unfortunately, bridges falls because of numbers, not deductions... its unfair considering the effort and fun of the seconds over the others.

Second thought, I used a lot in the past, the term "candidate solution" to describe my solutions. I used in the sense it is very easy myself to be wrong, but there is another meaning to the word "candidate" which happens to be very apt to be applied here: candidates are elected, and so, are solutions too. If by chance, everyone is wrong in the same direction, we might end be convicted that some wrong solution is the right solution. Although this might appear farfetched to some, it is not a big deal and is quite common. That's why we have scientific revolutions for a chance. No wonder, voting systems are a common tool in some algorithms used in the automatic learning community (artificial intelligence, knowledge databases, faulty detection systems, statistical learning, etc).

Cheers,
JPA

[onta...@hotmail.com]

[onta...@hotmail.com]

Hello Joao,
Thank you for your rather lengthy observations. In a nutshell, it is true that there is a different "mind-set" between those who like to compose puzzles, and those who only try to solve them. I think the differences speak for themselves. However the interaction between these two thinkers makes for good brain exercise and discussions.
Regards,
Bill.