Mixing
Cross sections of randomly distributed defects
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A $10 times 10 times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how many portions of spheres could be counted?
cross-sections
edited Aug 3 at 1:05
David G. Stork
7,3202728
asked Aug 3 at 1:03
James S
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up vote
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down vote
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A $10 times 10 times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how many portions of spheres could be counted?
cross-sections
edited Aug 3 at 1:05
David G. Stork
7,3202728
asked Aug 3 at 1:03
James S
1
2
Is 2-micron the radius or the diameter of the sphere? Is a cross-section found intersecting a random plane and the cube?
– Giulio Scattolin
Aug 3 at 1:37
1
The odds that a random chord in a unit circle has length less than sqrt(3) is either 1/2, 1/3, or 1/4, as shown in the random chord paradox. The random cross section is worse.
– Ed Pegg
Aug 3 at 2:06
@EdPegg For that paradox it is a matter of choosing the method for drawing the chords. By the way I think the "random cross section" calculus could be simplified observing that any area obtained by the intersecting a plane and a sphere (provided this is the method intended) is a circle whose radius can be desumed from the distance between the section and the center of the sphere.
– Giulio Scattolin
Aug 3 at 9:11
Lets say it's the radius
– James S
Aug 4 at 11:20
@JamesS It would be helpful if you defined how a "random cross-section" was done. As Ed Pegg noted, different ways in cutting the cube could yield different values for the probability of what you're asking.
– Giulio Scattolin
Aug 4 at 12:33
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A $10 times 10 times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how many portions of spheres could be counted?
cross-sections
edited Aug 3 at 1:05
David G. Stork
7,3202728
asked Aug 3 at 1:03
James S
1
A $10 times 10 times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how many portions of spheres could be counted?
cross-sections
edited Aug 3 at 1:05
David G. Stork
7,3202728
asked Aug 3 at 1:03
James S
1
edited Aug 3 at 1:05
David G. Stork
7,3202728
edited Aug 3 at 1:05
David G. Stork
7,3202728
edited Aug 3 at 1:05
David G. Stork
7,3202728
7,3202728
asked Aug 3 at 1:03
James S
1
asked Aug 3 at 1:03
James S
1
asked Aug 3 at 1:03
James S
1
1
2
Is 2-micron the radius or the diameter of the sphere? Is a cross-section found intersecting a random plane and the cube?
– Giulio Scattolin
Aug 3 at 1:37
1
The odds that a random chord in a unit circle has length less than sqrt(3) is either 1/2, 1/3, or 1/4, as shown in the random chord paradox. The random cross section is worse.
– Ed Pegg
Aug 3 at 2:06
@EdPegg For that paradox it is a matter of choosing the method for drawing the chords. By the way I think the "random cross section" calculus could be simplified observing that any area obtained by the intersecting a plane and a sphere (provided this is the method intended) is a circle whose radius can be desumed from the distance between the section and the center of the sphere.
– Giulio Scattolin
Aug 3 at 9:11
Lets say it's the radius
– James S
Aug 4 at 11:20
@JamesS It would be helpful if you defined how a "random cross-section" was done. As Ed Pegg noted, different ways in cutting the cube could yield different values for the probability of what you're asking.
– Giulio Scattolin
Aug 4 at 12:33
 |Â
show 1 more comment
2
Is 2-micron the radius or the diameter of the sphere? Is a cross-section found intersecting a random plane and the cube?
– Giulio Scattolin
Aug 3 at 1:37
1
The odds that a random chord in a unit circle has length less than sqrt(3) is either 1/2, 1/3, or 1/4, as shown in the random chord paradox. The random cross section is worse.
– Ed Pegg
Aug 3 at 2:06
@EdPegg For that paradox it is a matter of choosing the method for drawing the chords. By the way I think the "random cross section" calculus could be simplified observing that any area obtained by the intersecting a plane and a sphere (provided this is the method intended) is a circle whose radius can be desumed from the distance between the section and the center of the sphere.
– Giulio Scattolin
Aug 3 at 9:11
Lets say it's the radius
– James S
Aug 4 at 11:20
@JamesS It would be helpful if you defined how a "random cross-section" was done. As Ed Pegg noted, different ways in cutting the cube could yield different values for the probability of what you're asking.
– Giulio Scattolin
Aug 4 at 12:33
2
2
Is 2-micron the radius or the diameter of the sphere? Is a cross-section found intersecting a random plane and the cube?
– Giulio Scattolin
Aug 3 at 1:37
Is 2-micron the radius or the diameter of the sphere? Is a cross-section found intersecting a random plane and the cube?
– Giulio Scattolin
Aug 3 at 1:37
1
1
The odds that a random chord in a unit circle has length less than sqrt(3) is either 1/2, 1/3, or 1/4, as shown in the random chord paradox. The random cross section is worse.
– Ed Pegg
Aug 3 at 2:06
The odds that a random chord in a unit circle has length less than sqrt(3) is either 1/2, 1/3, or 1/4, as shown in the random chord paradox. The random cross section is worse.
– Ed Pegg
Aug 3 at 2:06
@EdPegg For that paradox it is a matter of choosing the method for drawing the chords. By the way I think the "random cross section" calculus could be simplified observing that any area obtained by the intersecting a plane and a sphere (provided this is the method intended) is a circle whose radius can be desumed from the distance between the section and the center of the sphere.
– Giulio Scattolin
Aug 3 at 9:11
@EdPegg For that paradox it is a matter of choosing the method for drawing the chords. By the way I think the "random cross section" calculus could be simplified observing that any area obtained by the intersecting a plane and a sphere (provided this is the method intended) is a circle whose radius can be desumed from the distance between the section and the center of the sphere.
– Giulio Scattolin
Aug 3 at 9:11
Lets say it's the radius
– James S
Aug 4 at 11:20
Lets say it's the radius
– James S
Aug 4 at 11:20
@JamesS It would be helpful if you defined how a "random cross-section" was done. As Ed Pegg noted, different ways in cutting the cube could yield different values for the probability of what you're asking.
– Giulio Scattolin
Aug 4 at 12:33
@JamesS It would be helpful if you defined how a "random cross-section" was done. As Ed Pegg noted, different ways in cutting the cube could yield different values for the probability of what you're asking.
– Giulio Scattolin
Aug 4 at 12:33
 |Â
show 1 more comment
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2
Is 2-micron the radius or the diameter of the sphere? Is a cross-section found intersecting a random plane and the cube?
– Giulio Scattolin
Aug 3 at 1:37
1
The odds that a random chord in a unit circle has length less than sqrt(3) is either 1/2, 1/3, or 1/4, as shown in the random chord paradox. The random cross section is worse.
– Ed Pegg
Aug 3 at 2:06
@EdPegg For that paradox it is a matter of choosing the method for drawing the chords. By the way I think the "random cross section" calculus could be simplified observing that any area obtained by the intersecting a plane and a sphere (provided this is the method intended) is a circle whose radius can be desumed from the distance between the section and the center of the sphere.
– Giulio Scattolin
Aug 3 at 9:11
Lets say it's the radius
– James S
Aug 4 at 11:20
@JamesS It would be helpful if you defined how a "random cross-section" was done. As Ed Pegg noted, different ways in cutting the cube could yield different values for the probability of what you're asking.
– Giulio Scattolin
Aug 4 at 12:33