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Hole in a sphere


Best Answer bonanova, 23 August 2007 - 07:11 AM

The volume of the spherical caps is given by:
[list]
where
[list]
[*] h = the height of the cap (difference between r and the distance from the centre of the[/*:m:1cc31][list] sphere to the centre of the circular end of the hole)


Kudos to cpotting for the cap formula.
Spoiler for Here's the mathematical solution
Spoiler for Here's the logical solution
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166 replies to this topic

#1 bonanova

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Posted 21 August 2007 - 03:17 PM

Maybe this has already been posted. A friend asked me this a while back, and I answered her in less than a minute.
She said I was a genius. But I said there were two ways to arrive at the answer, and I simply chose the easier way.

A 6-inch
[long] hole is drilled through [the center of] a sphere.
What is the volume of the remaining portion of the sphere?

The hard way involves calculus. The easy way uses logic.

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
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#2 Riddari

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Posted 21 August 2007 - 11:12 PM

Well,I must not be a genius because the easy way is not jumping in my head. The difficult way would be to determine the volume of the cut out section and deduct it from the volume of the sphere. I am too lazy to do all the work actually involved with doing that though.

It does seem to me that it would be fairly important to know the orientation of the hole. It is natural to assume the hole goes directly through the center of the sphere, but that is really not stated in the question. If I am not mistaken, the problem is a lot more difficult if the hole is off center.

Please post at least the easy answer. I really want to know if I should kick myself for not seeing it immediately.
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#3 Aaron Burr

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Posted 22 August 2007 - 03:08 AM

I'm going out on a limb, saying it's more of a lateral thinking puzzle and the volume is none(?). If you drill a whole though a sphere then it is no longer a sphere (at least in the technical sense that i learned back in geometry).

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#4 Martini

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Posted 22 August 2007 - 03:36 AM

Please post at least the easy answer.


IMO, riddles are more fun when posters get to mull over solutions without the OP providing an answer so soon after posting the riddle. But this is bonanova's riddle and he can do as he pleases; I'm just adding my 2 cents.

I'm going out on a limb, saying it's more of a lateral thinking puzzle and the volume is none(?). If you drill a whole though a sphere then it is no longer a sphere (at least in the technical sense that i learned back in geometry).


No, it's not a lateral thinking puzzle and it doesn't matter if a sphere with a hole can still be called a sphere or not.

To clarify, I believe bonanova is speaking of a hole 6 inches long of an unspecified diameter. Am I correct bonanova?
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#5 jpaterson3

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Posted 22 August 2007 - 03:45 AM

If you take the meaning of volume of the sphere as the amount of space it takes up then its volume would be the same. If you take volume as the amount of material that makes up the sphere I have no idea
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#6 bonanova

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Posted 22 August 2007 - 09:55 AM

Clarifications:

[1] the hole is a circular cylinder of empty space whose axis passes through the center of the sphere - just as a drill would make if you aimed the center of the drill at the center of the sphere and made sure you drilled all the way through.

[2] the length of the hole [6 inches] is the height of the cylinder that forms the inside surface once the hole is drilled. picture the inside surface as viewed from inside the hole and measure the length of that surface in the direction of the axis of the drill.

in this sense, you could for example drill a 6-inch hole through the earth. the diameter of the hole would be huge, and you'd just have a tiny remnant of the earth left. but if you could set it on a table [a big table] it would be 6 inches high.

you of course could not drill a 6-inch hole through a sphere whose diameter was less than 6 inches. it was actually this fact that led me to the logical answer and made me a genius for a couple of minutes.
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#7 bonanova

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Posted 22 August 2007 - 10:02 AM

To clarify, I believe bonanova is speaking of a hole 6 inches long of an unspecified diameter. Am I correct bonanova?


Yes.
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#8 bonanova

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Posted 22 August 2007 - 10:15 AM

Well,I must not be a genius because the easy way is not jumping in my head. The difficult way would be to determine the volume of the cut out section and deduct it from the volume of the sphere. I am too lazy to do all the work actually involved with doing that though.


I did the calculus afterward, and however you do the calculation, it's difficult. The remaining volume is a volume of revolution with requires finding the cross sectional area and spinning it thru 360 degrees. It's no easier to compute the cylindrical volume removed, cuz there are spherical "caps" on the cylinder which don't have formulas that I could find.

The logical way is easier, and I'll post it in a day or so if you don't get it... have fun.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

#9 cpotting

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Posted 22 August 2007 - 11:54 PM

It's no easier to compute the cylindrical volume removed, cuz there are spherical "caps" on the cylinder which don't have formulas that I could find.


The volume of the spherical caps is given by:
[list]
where
[list]
[*] h = the height of the cap (difference between r and the distance from the centre of the[/*:m:79969][list] sphere to the centre of the circular end of the hole)

This still doesn't make it easy to calculate
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#10 Writersblock

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Posted 23 August 2007 - 06:46 AM

I may be missing something here, as I was a Literature Major not a Math guy, but wouldn't you need 1 more piece of info regardless of semantics?

If you mean by "a six inch hole through a sphere" a six inch long hole "through" the sphere(all the way), then you need to know the diameter of the hole. We know the diameter of the sphere is 6 inches plus pi * h*h * (r - h / 3) , but the hole could be from microscopic to almost 6 inches plus pi * h*h * (r - h / 3) in diameter; thus, size matters.

If you mean by "a six inch hole through a sphere" a six inch diameter hole through a sphere of unknown size, then you need to know the diameter of the sphere. It can be from a little over 6 inches to some outrageously large finite number; thus, size matters.

The moral of the story is: your girlfriend lied. Size DOES matter. ;)

But seriously, if the clarification above, in fact, means that it's a 6 inch hole THROUGH a sphere, we know only that the sphere must be 6 inches plus pi * h*h * (r - h / 3) in diameter, or one of two things are incorrect: either the hole is NOT 6 inches long, or it's not THROUGH the sphere - it is merely INTO the sphere. But we still don't know the diameter of the hole. IF it is say 5.9" in diameter, then we've made a really cool hat and can figure out the volume. If it's, say, 1" in diameter then we have a giant bead on our hands and can figure out the volume. But without this extra piece of knowledge, the answer set, HAS to be somewhere between > 0 and 4/3 pi times (6 inches plus pi * h*h * (r - h / 3)) cubed.
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