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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
Go to the full post

166 replies to this topic

### #131 NickFleming

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Posted 21 July 2010 - 05:22 PM

No no no, circumference wouldn't vary with size of original object. Volume, however, definitely would. Circumference is the distance around a circle. And your spoiler just doesn't make sense to me.

Edited by NickFleming, 21 July 2010 - 05:23 PM.

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### #132 NickFleming

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Posted 21 July 2010 - 05:29 PM

Okay, we're missing two factors we need; The original volume, and the diameter/circumference of the hole. So I'll just write it in terms of the two.
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### #133 NickFleming

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Posted 21 July 2010 - 05:41 PM

First, find volume of original sphere. volume of sphere=2/3 volume of smallest cylinder it could have been cut from. Assuming 6 inches is all the way through the hole, the diameter of sphere, and therefore the original cylinder, is 6 inches. so to find volume of cylinder, we set up pi times 32 times 6. so then pi times 54. I'll just leave it in terms of pi to be most exact. So, Archimedes found out that volume of sphere=2/3 that of cylinder. So now we have 32pi=volume of sphere. So answer would be 32p-(6pr^2), p being pi, and r representing missing radius of hole. This is because missing volume would be much different depending on whether diameter of hole was 6m or 600.

Restatement of answer: 32p - 6pr2

Any problems with my logic, tell me. And yes, I kno I found volume of sphere different than most people would. Turned out we didn't need original volume.

Edited by NickFleming, 21 July 2010 - 05:42 PM.

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### #134 mojobrain

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Posted 21 July 2010 - 10:23 PM

Spoiler for Maybe I'm missing something here, was never good in math, but:

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### #135 t8t8t8

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Posted 22 July 2010 - 01:05 AM

I think I understand what they meant, but I could be wrong.

In the picture, the holes are both 6 inches long(the dark gray part), but there are caps (the light gray part) that are cut off.
So only the white part is left and that's the part we are looking for.
That's why they don't tell you the diameter. It changes depending on the size of the sphere.
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### #136 Arbelle

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Posted 22 July 2010 - 09:31 AM

First, find volume of original sphere. volume of sphere=2/3 volume of smallest cylinder it could have been cut from. Assuming 6 inches is all the way through the hole, the diameter of sphere, and therefore the original cylinder, is 6 inches. so to find volume of cylinder, we set up pi times 32 times 6. so then pi times 54. I'll just leave it in terms of pi to be most exact. So, Archimedes found out that volume of sphere=2/3 that of cylinder. So now we have 32pi=volume of sphere. So answer would be 32p-(6pr^2), p being pi, and r representing missing radius of hole. This is because missing volume would be much different depending on whether diameter of hole was 6m or 600.

Restatement of answer: 32p - 6pr2

Any problems with my logic, tell me. And yes, I kno I found volume of sphere different than most people would. Turned out we didn't need original volume.

I didn't understand all of your answer, but I have a small remark. 6 inches is the hight of the hole's "wall", not neccesariy the diameter of the sphere. Actually, if the size of the hole was zero, the sphere would be 6 inches diameter.

Hope I uploaded my sketch correctly. I know t8t8t8 beat me to it, but I was really excited drawing it last night...

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### #137 NickFleming

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Posted 22 July 2010 - 03:18 PM

I didn't understand all of your answer, but I have a small remark. 6 inches is the hight of the hole's "wall", not neccesariy the diameter of the sphere. Actually, if the size of the hole was zero, the sphere would be 6 inches diameter.

Hope I uploaded my sketch correctly. I know t8t8t8 beat me to it, but I was really excited drawing it last night...

Maybe I messed up, I don't know. You may have been confused about sphere volume thing, look up Archimedes on wikipedia. I think it was volume, but maybe it was surface area. I got the impression that the hole is drilled 6 inches into the sphere, and the 'walls' of a cylindrical hole are curved.
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### #138 lcornea

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Posted 17 August 2010 - 02:15 PM

There are simpler ways to do this.

Spoiler for The easyest solution

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### #139 InvincibleRam

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Posted 17 October 2010 - 09:38 PM

This is my first post..
So I invite myself in...

Spoiler for

I once heard a puzzle of having a steel wire that ties around the earth, then we add one meter of wire to it. how much would the earlier wire rise above the ground.

The answer is that it is all standard, irrespective whether it is an orange or the Earth.
:-)

Edited by InvincibleRam, 17 October 2010 - 09:41 PM.

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### #140 Tsopi

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Posted 18 October 2010 - 07:41 PM

How can you possibly know the answer if you don't know how wide the whole is?
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