The problem as originally stated is not solvable, as it could be reasonably interpreted as a 6-inch diameter hole all the way through an object.
I would argue that you cannot drill a 6 inch hole through a sphere with radius = 3 any more than you can through a sphere of r = 2. A hole with no volume is not a hole. It may be true that the r=3 case is a boundry which a solution approaches, and obviously the volume of such an intact sphere is 4/3 * 3 * 3 * 3 * pi cu. in.= 36 pi cubic inches.
There was also discussion of the 'caps' that would fit on top of the inscribed cylinder in the problem. Clearly, for the example of boring through the earth with a bit whose radius is almost as big as the radius of the earth, these caps would be massive (basically, you've just cut away 6-inch slice of the earth through the center). The volume of these caps is very obviously dependent on the radius of the sphere, and are thus not included in the 'answer'.
There was another purported 'solution' which seems to me to be flawed:
You're all making this too hard. You just need to know the area of each slice of the donut, which is simply the difference of two circles.
L = length of bore
r = radius of bore
R = radius of sphere
x = radius of sphere cross-section at height z
A = area of donut cross-section at height z
Forming two simple right triangles:
r? = R? - L?/4
x? = R? - z?
The first equation stipulates that the radius of the cylinder equals the radius of the sphere minus (the length of the bore divided by 4), which means that
r = R - 1.5 which could be true for one such case, but is hardly a general solution - if the difference between the radii of the cylinder and sphere were 1.5 inches then the volume would be way more than 36 pi cubic inches.
Also, the stated 'solution' of 36pi isn't a volume at all, whereas 36 pi cubic inches is...
But still, great for you that you felt like a genius and thanks for sharing.
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.
... combining 2 posts by the author ...
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.
