markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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Draw four dots that mark the corners of a perfect square as shown in the illustration.
The object is to draw a minimal network spanning them. The parts of it may intersect, and you're allowed to use additional dots while drawing the network.
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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A cylindrical hole six inches long has been drilled straight through the center of a solid sphere - just as shown in the illustration.
The question is what is the volume remaining in the sphere?
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JohnD
Reged: Oct 21 2002
Posts: 46
Loc: Chicago Suburbs
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I am not sure what you mean by "network". By that, do you mean that each node has a DIRECT connection to every other node? Or, that there is a path that can be taken such that you can eventually get from one node to another? (P.S. The latter is 3 lines.)
Also--on that sphere: Note that the radius of the sphere and the diameter of the hole are not given! I am starting to do the math, but it is a bit "interesting" since I must use unknowns all the way through, and that always got me in trouble at school as I would invariably make a typographical error.
Edited by JohnD (Tue Jun 01 2004 05:44 PM)
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Luka
Reged: Apr 25 2002
Posts: 1401
Loc: The great NorthWet
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The latter would be two lines.
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The true measure of a man is how he treats someone who can do him absolutely no good. ~Samuel Johnson
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JohnD
Reged: Oct 21 2002
Posts: 46
Loc: Chicago Suburbs
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Luka--if you are talking about two diagonal lines, with a node added where they cross, we can argue that that is actually FOUR lines. When I design that sort of system, I consider the connection from one node to another to be one line.
But we have not heard anything from Mr. Cadioli who should be adding some sort of devious hints from time to time. Where is that fine gent from Oz with the strine accent?
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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John
A direct connection is not nessesarily the shortest route
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JohnD
Reged: Oct 21 2002
Posts: 46
Loc: Chicago Suburbs
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Ah, what you are saying is that the TOTAL LENGTH of the lines that you draw must be the shortest possible.
Hmmm........Now for the fun part.
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Luka
Reged: Apr 25 2002
Posts: 1401
Loc: The great NorthWet
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You are adding rules where they weren't originally.
The dots aren't nodes. The lines are the network. Period. You CAN add extra dots, but why ? Where does it say you need a dot, (or "node"), to turn the corner ? You turn where the lines intersect.
It says intersect, not cross, or overlap, etc...
Railroads and rivers cross. Streets intersect.
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The true measure of a man is how he treats someone who can do him absolutely no good. ~Samuel Johnson
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Luka
Reged: Apr 25 2002
Posts: 1401
Loc: The great NorthWet
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Here's the original solution.
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The true measure of a man is how he treats someone who can do him absolutely no good. ~Samuel Johnson
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Luka
Reged: Apr 25 2002
Posts: 1401
Loc: The great NorthWet
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Heres a solution using three lines, and giving minimal travelling distance between the four dots...
Note the red line does not quite touch any of the dots.
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The true measure of a man is how he treats someone who can do him absolutely no good. ~Samuel Johnson
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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sorry..that's not the shortest way ( but you are getting warm )
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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nope..( very cold )
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SamT
Reged: Jun 05 2004
Posts: 2
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>>The question is what is the volume remaining in the sphere?
All of it.
In mathematics, a sphere is a quadric consisting only of a surface and is therefore hollow. In non-mathematical usage a sphere is often considered to be solid; mathematicians call this the interior of the sphere. 
SamT
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Luka
Reged: Apr 25 2002
Posts: 1401
Loc: The great NorthWet
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Sam, the wording is, how much volume remains IN the sphere.
Not OF the sphere...
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The true measure of a man is how he treats someone who can do him absolutely no good. ~Samuel Johnson
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kostello
Reged: Nov 18 2003
Posts: 27
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i'm having a hard time trying to calculate the diameter of the cylinder.
i have found all the formulae for the rest of it but i need the cylinder diameter to get the diameter of the sphere using pythagoras.
give me a hint.
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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Kostello
here is a hint for you. the residue is constant regardless of the hole's diameter or the size of the sphere
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Luka
Reged: Apr 25 2002
Posts: 1401
Loc: The great NorthWet
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Well then, SamT must be right.
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The true measure of a man is how he treats someone who can do him absolutely no good. ~Samuel Johnson
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kostello
Reged: Nov 18 2003
Posts: 27
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sooooo
i've figured it out, but i had to do some googling to find out how and to check the answer.
for those of you still perplexed the answer doesn't change with the radius of the hole.
so what you need to do is pick a value for the radius and calculte away.
i spent ages reading several proofs of how to calculate it and i must say my brain did start to hurt a bit.
good exercise though.
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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yes kostello did provide the correct answer. an interesting exercise. I'll post the solution this coming weekend for any who are interested.
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markcadioli
Reged: Apr 25 2002
Posts: 1406
Loc: Australia
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Hole in the Sphere
(solution)
If you want to avoid the calculations in Solution 1, simply take a look directly at Solution 2 below it.
Solution 1
Let R be the radius of the sphere. As the illustration indicates, the radius of the cylindrical hole will then be the square root of R2 - 9, and the altitude of the spherical caps at each end of the cylinder will be R - 3. To determine the residue after the cylinder and caps have been removed, we add the volume of the cylinder, 6?(R2 - 9), to twice the volume of the spherical cap, and subtract the total from the volume of the sphere, 4?R3/3. The volume of the cap is obtained by the following formula, in which A stands for its altitude and r for its radius: ?A(3r2 + A2)/6.
When this computation is made, all terms obligingly cancel out except 36? - the volume of the residue in cubic inches. In other words, the residue is constant regardless of the hole's diameter or the size of the sphere!
Solution 2
John W. Campbell, Jr., editor of Astounding Science Fiction, was one of several readers who solved the sphere problem quickly by reasoning adroitly as follows: The problem would not be given unless it has a unique solution. If it has a unique solution, the volume must be a constant which would hold even when the hole is reduced to zero radius. Therefore the residue must equal the volume of a sphere with a diameter of six inches, namely 36?.
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