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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

### #61 roolstar

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Posted 22 February 2008 - 09:52 AM

Does this help?

I think this is exactly what Bonanova was describing...

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### #62 rvawter

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Posted 24 February 2008 - 06:04 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 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.

Hey, how bought you post the answer to the easy way? its been driving me nuts for about a week
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### #63 bonanova

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Posted 24 February 2008 - 06:24 AM

Hey, how bought you post the answer to the easy way? its been driving me nuts for about a week

The easy way is to suppose the answer is the same for
any sphere [with diameter not less than 6 inches], and
calculate the answer for a 6-inch diameter sphere.

The answer is 36pi cubic inches - exactly the volume of the sphere.

The hard way, and why the easy way works, can be found earlier
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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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### #64 storm

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Posted 24 February 2008 - 10:25 AM

This is a hell of a puzzle. The problem is most people didnot understand the problem!!!
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### #65 roolstar

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Posted 24 February 2008 - 02:42 PM

So we can visualize an infinitely elastic sphere of 6 inch in diameter.

If we make a cylinder with variable diameter go through it (the cylinder can get wider and wider), and increase its diameter to any value we want: the sphere will stretch around the cylinder and will remain 6 inchs tall!

The volume of the sphere will not change but the sphere will become more & more flattened around the cylinder...

WOW

Edited by roolstar, 24 February 2008 - 02:43 PM.

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### #66 Jkyle1980

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Posted 25 February 2008 - 04:13 AM

This may help those that are not quite seeing how the hole is being drilled. Imagine instead that you are drilling a hole into the top of a mountain. You have a drill that is 4 ft. wide. When you begin, the drill is wider than the very top of the mountain so all dirt is removed. If you stopped at this point, you wouldn't have a hole on top, you'd just have a mountain with a flat top. You have to keep drilling until the diameter of the cross-section of the mountain at the point your drill has reached is greater than 4 ft. (diameter of the drill) before you begin actually making a hole. See the picture below. It is not to scale, and I made it in PowerPoint when I should have been working so cut me some slack.

Now imagine that you were drilling through a sphere instead. Same idea of the hole not starting until there is matter left on the outside of the drill and stopping as soon as the drill busted through the bottom. This is the easiest way for me to see it.

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Posted 02 March 2008 - 12:57 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 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.

Thinking of it differently, the cylinder expands to the sphere, right? So, at any time a hole is drilled through the sphere, the remaining portion always measures 6 inches in height. Therefore, a 6-inch sphere cannot be drilled and must be the "ideal" volume. Therefore the volume would be 36*pi, right?
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### #68 bonanova

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Posted 02 March 2008 - 11:26 PM

Thinking of it differently, the cylinder expands to the sphere, right? So, at any time a hole is drilled through the sphere, the remaining portion always measures 6 inches in height. Therefore, a 6-inch sphere cannot be drilled and must be the "ideal" volume. Therefore the volume would be 36*pi, right?

Bingo!
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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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Posted 03 March 2008 - 12:07 AM

you have to know some type of measurment besides a 6in hole
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### #70 bonanova

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Posted 03 March 2008 - 05:24 AM

you have to know some type of measurment besides a 6in hole

It certainly seems so at first glance. But are you certain?
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

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