# Hole through sphere



## robert@fm (Sep 15, 2018)

A hole is drilled precisely through the centre of a sphere. When measured, it is found to be exactly six inches long.

What volume of material remains in the sphere? (Surprisingly, there _is_ enough data to work out the answer, although it doesn't look like it.)


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## Sally71 (Sep 15, 2018)

None, because it all fell out of the hole!


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## robert@fm (Sep 15, 2018)

^Nope, it's a solid sphere (made of wood or the like) It isn't hollow either.


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## Hepato-pancreato (Sep 16, 2018)

What about the size of the drill bit.


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## Drummer (Sep 16, 2018)

Well - the volume of a sphere is 1/6th of pi times the diameter cubed - so it would be pi times 36 to start with - 113.097 cubic inches - assuming that the length of the hole is not less than the diameter of the sphere due to it cutting away the curve - but it would surely be reduced by a volume of 6 times pi r squared, where the r is the radius of the drill.Wouldn't it? And we do not know r.


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## silentsquirrel (Sep 16, 2018)

"exactly"???  I would quibble with being able to measure anything exactly, depends on the accuracy of the measuring tool.....    Is it 6 rather than 5 or 7, or 6.0 rather than 5.9 or 6.1, etc, etc?  Many of us  intuitively expect our meters to be accurate to nearest 0.1 because they read to 1 dp, however much we are told they are not that accurate!
How do you measure the length of a hole?   Centre to centre is unlikely to be very accurate, unless it is a very small radius hole.  (Perhaps answer is 100% as we must consider the radius to be so small it is negligible, to fit with that "exactly"!).  I think one would measure along the edge of the hole, so radius of the sphere is a bit more than 3.  Hmmm.....  I'll maybe have another think later, got to go out now.


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## Eddy Edson (Sep 16, 2018)

As far as I can see, it's going to depend on the radius of the sphere - not the width of the hole, which will be a function of the sphere's radius and the 6 inch length of the hole. But I'm probably missing something.


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## Eddy Edson (Sep 16, 2018)

OK: 36 times _pi
_
If that's not right I've just wasted 30 min of my life. Or even if it is, really. 

EDIT: If you've forgotten the formula for the volume of a spherical cap  https://en.wikipedia.org/wiki/Spherical_cap


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## robert@fm (Sep 16, 2018)

@Eddy Edson has got it right! (Incidentally, a hole drilled through the centre of a sphere is a cylinder; the length is the height of the cylinder.)

It is possible to work this one out by assuming an arbitrary radius for the sphere, then calculating (as a formula) how wide the hole must be in order to be six inches long, devising formulae for the volume of the hole and of the end-caps removed by drilling it, and subtracting those amounts from the volume of the sphere; if you do that, just about everything cancels except 36*pi, the volume of what remains of the sphere (in cubic inches).

The easier way to solve it is to realise that this is a puzzle, and wouldn't have been given if it didn't have a unique solution; one which must hold even if the radius of the hole is zero, in which case the volume of the residue of the sphere is that of a sphere of three inches radius, namely 36*pi cubic inches.


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## mikeyB (Sep 16, 2018)

Blimey, if you are digging up old Martin Gardener puzzles, here’s another: 

Two trains are travelling at 30 miles an hour - towards each other before a horrible collision. They are two miles apart. A very energetic fly flies from the front of one train to the front of the other, and repeats this back and forth until they crash.

How far has it flown before it gets squished? Assume no acceleration and deceleration.


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