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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

### #141 bonanova

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Posted 18 October 2010 - 08:39 PM

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

That's what makes this a great puzzle.

Spoiler for Here's the easy solution

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### #142 maurice

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Posted 18 October 2010 - 09:24 PM

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

Sure you do. Its radius is (R^2-9)^(1/2) where R is the radius of the sphere.
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### #143 maurice

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Posted 18 October 2010 - 09:43 PM

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

Sure you do. Its radius is (R^2-9)^(1/2) where R is the radius of the sphere.

Or if by "whole" you mean the entire sphere then its 2R.

Edited by maurice, 18 October 2010 - 09:45 PM.

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### #144 Asitaka

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Posted 19 October 2010 - 04:26 AM

After reading through this post, I don't understand how this can be 36pi for the volume. Let's say the sphere is 6 inches in diameter, and you pull out a drill that has the same diameter as the sphere and somehow magically you drill through it... now your sphere is just wood chips on the floor.... How much volume is remaining? 0, because you made the whole sphere a hole...
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### #145 bonanova

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Posted 19 October 2010 - 05:22 AM

After reading through this post, I don't understand how this can be 36pi for the volume. Let's say the sphere is 6 inches in diameter, and you pull out a drill that has the same diameter as the sphere and somehow magically you drill through it... now your sphere is just wood chips on the floor.... How much volume is remaining? 0, because you made the whole sphere a hole...

Six inches in the Length, not the Diameter, of the hole.

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### #146 phaze

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Posted 19 October 2010 - 05:47 AM

Confusion comes about as it is not exactly defined what the 6 inches drilled through is a measure of

I think this might explain what bonanova is asking

Lets just say for example that the sphere was the size of the earth the drill bit diameter is of such a width that by the time the earth has been drill through you have a doughnut 6 inches high. If this is the earth really thin sides with a massive hole in the middle and the drill has drilled a lot more than six inches. If the sphere is 6 inches high the drill bit must be 0 inches in diameter for the sphere to remain 6 inches high.

in other words this ( ) is 6 inches this O is not necessarily 6 inches
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### #147 Asitaka

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Posted 19 October 2010 - 07:57 AM

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 a sphere.
What is the volume of the remaining portion of the sphere?

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

Quoting the original post, it says a 6 inch long hole, it doesn't specify the width.

I assume since it says 'through' the sphere, the hole must connect two outer edges of the sphere, be it straight through the center or off-centered or on a diagonal.

With only this information, I could theoretically get a drill bit whose diameter is the same as the sphere's diameter (or larger) and drill the sphere to nothingness no matter where the 6 inch long hole was drilled through it.

Google 'World's largest drill bit'. That would make a BIG difference compared to your standard hand held drill.
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### #148 maurice

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Posted 19 October 2010 - 07:52 PM

Quoting the original post, it says a 6 inch long hole, it doesn't specify the width.

I assume since it says 'through' the sphere, the hole must connect two outer edges of the sphere, be it straight through the center or off-centered or on a diagonal.

With only this information, I could theoretically get a drill bit whose diameter is the same as the sphere's diameter (or larger) and drill the sphere to nothingness no matter where the 6 inch long hole was drilled through it.

Google 'World's largest drill bit'. That would make a BIG difference compared to your standard hand held drill.

If you had a drill bit whose diameter was the same as that of the sphere you would not be able to drill a 6 inch hole...so no you can't have the diameter be the same. The width of the hole is not specified but it is implied as the only possible radius of the hole that will generate a 6 inch hole is (R^2-9)^(1/2) where R is the radius of the sphere.

Edited by maurice, 19 October 2010 - 07:53 PM.

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### #149 phaze

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Posted 19 October 2010 - 09:15 PM

With only this information, I could theoretically get a drill bit whose diameter is the same as the sphere's diameter (or larger) and drill the sphere to nothingness no matter where the 6 inch long hole was drilled through it.

If it was a 6 inch high cylinder you might have better luck. You would still have to successfully argue that nothing constitutes a 6 inch hole with infinitly thin sides. As it is a sphere if the drill is the same size as the sphere the hole at the end of drilling is 0 inches (the drill bit may have traveled 6 inches or more but we are measuring the hole not drill travelling). The depth of the hole (or height of the resulting doughnut) changes when you drill through the sphere as the drill knocks off the top and bottom of the sphere.

Although we are not told the width of the sphere or the width of the drill the width of sphere and drill are related. With a sphere with a 6 inch diameter the drill must be 0 inches wide, with a 60 inch diameter sphere the drill will need to be about 59.7 inches (the drill will travel 60 inches, but the remaining hole will be 6 inches)
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### #150 wolfgang

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Posted 21 October 2010 - 09:08 PM