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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

### #41 bonanova

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Posted 06 February 2008 - 02:59 PM

Length.
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### #42 Jkyle1980

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Posted 07 February 2008 - 06:43 AM

So the hole is 6 inches deep? And so how does putting a hole 6 inches deep in the earth make the volume of the remaining earth 36pi?
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### #43 ClattoVerataNicto

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Posted 07 February 2008 - 07:01 AM

As long as you told us what the radius of the sphere was then any fool could tell you that your Volume was:

4/3 ¶ r³ - ¶ 6² r ! DUH
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### #44 Jkyle1980

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Posted 07 February 2008 - 08:05 AM

As long as you told us what the radius of the sphere was then any fool could tell you that your Volume was:

4/3 ¶ r³ - ¶ 6² r ! DUH

Who is this person that speaks to me as if I needed his (or her) advice?
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### #45 bonanova

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Posted 07 February 2008 - 10:00 AM

As long as you told us what the radius of the sphere was then any fool could tell you that your Volume was:

4/3 ¶ r³ - ¶ 6² r ! DUH

A moment's reflection shows the fool's answer is incorrect.
Spoiler for proof

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### #46 bonanova

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Posted 07 February 2008 - 10:33 AM

So the hole is 6 inches deep? And so how does putting a hole 6 inches deep in the earth make the volume of the remaining earth 36pi?

After the hole is drilled completely through the sphere, not just part way into the sphere,
the height[*] of the remaining part of the sphere is 6 inches.
[*] assuming the drill was vertical.

That's what's meant by drilling a 6" [long] hole thru a sphere.

If the sphere was the earth, the drill's diameter must have been just less than the earth's diameter,
leaving a 6" wide bracelet of earth that includes the equator and 3" either side of it.

In the above quote, visualize the drill that went only 6" into the earth.

Now use that drill to go all the way through the earth.
Measure the length of that hole.
If it's not 6", the drill has the wrong diameter.

I'm estimating that sometime, perhaps around Valentine's Day, the sound of a forehead slap will echo through the hole.
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### #47 Freerefill

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Posted 07 February 2008 - 02:10 PM

I didn't really understand it until this post.. but now I do.

Going back to the calculus answer, in order to find the volume, you'd have the sphere, the cylinder, and two end caps. Adding the cylinder and the end caps gives you the total volume removed from the sphere.

The condition is that the cylinder must be 6 inches long. The diameter is not specified or necessary; it simply must be 6 inches long. Or high. Whichever variable makes you happy. Also, note that this is the cylinder not counting the end caps.

Consider a large sphere and a really, really large person wanting a 6 inch bracelet. You have a drill and various drill bits all infinitely long (or, for all practical purposes, long enough to drill through the sphere with room to spare) but with varying diameter (length isn't important; diameter is). In order to make that 6 inch bracelet, you would have to choose a drill bit with a diameter large enough such that when you drill a hole completely through the sphere (going through the center, of course), the cylinder that you took out (again, not counting the end caps) would be 6 inches long. Thus, when you lay what's left over (the bracelet) flat on a table, it will be 6 inches high.

That is, assuming, that the sphere is large enough to accomplish this.

Consider a sphere infinitesimally larger than 6 inches in diameter (take the limit as diameter approaches 6). If you wanted to drill a hole through the sphere such that the cylinder which is removed (or, alternatively, the bracelet which is left behind) is 6 inches high, well, the sphere is already nearly 6 inches high itself, so you would need an infinitesimally thin drill. At 6 inches exactly, there's no way you can drill a hole which, when calculating the volume removed, would be composed of a 6 inch cylinder and two end caps.

Just my two cents. Hope it helps
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### #48 monkeymirth

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Posted 07 February 2008 - 06:01 PM

Okay, I'm not good at Calculus, but the easiest way i see it is that you throw the remaining part of the sphere into a bucket of water and see how much volume is displaced. However, I'm assuming that this solid sphere has a density greater than water so that it'll sink enough for the volume displacement.
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### #49 Jkyle1980

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Posted 07 February 2008 - 11:24 PM

After the hole is drilled completely through the sphere, not just part way into the sphere,
the height[*] of the remaining part of the sphere is 6 inches.
[*] assuming the drill was vertical.

That's what's meant by drilling a 6" [long] hole thru a sphere.

If the sphere was the earth, the drill's diameter must have been just less than the earth's diameter,
leaving a 6" wide bracelet of earth that includes the equator and 3" either side of it.

In the above quote, visualize the drill that went only 6" into the earth.

Now use that drill to go all the way through the earth.
Measure the length of that hole.
If it's not 6", the drill has the wrong diameter.

I'm estimating that sometime, perhaps around Valentine's Day, the sound of a forehead slap will echo through the hole.

So this? (And that sound isn't the slapping of a forehead. It's of the pounding of a forehead against a desk. haha)

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### #50 bonanova

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Posted 08 February 2008 - 09:24 AM

That's the idea...
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