Math puzzle
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JuicyBurger
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Define "full".
Will it not be "half full" with air almost instantly? Albeit at a low pressure.
Will it not be "half full" with air almost instantly? Albeit at a low pressure.
mmm4444bot
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If a cylindrical hole is drilled through the center of a sphere (along the sphere's axis) such that the length of the cylindrical hole is 6 units, what is the drilled sphere's remaining volume?
Clarification: The remaining volume is a Real number of cubic units, not an algebraic expression with variable(s).
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Hint: No additional information is required, to answer this question.
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Sloppy sketch:
[attachment=0:1x61ry4a]Bead.JPG[/attachment:1x61ry4a]
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MY EDIT: Added clarification about Real-number answer, and then I came back and added it properly. :roll:
If a cylindrical hole is drilled through the center of a sphere (along the sphere's axis) such that the length of the cylindrical hole is 6 units, what is the drilled sphere's remaining volume?
Clarification: The remaining volume is a Real number of cubic units, not an algebraic expression with variable(s).
?
Hint: No additional information is required, to answer this question.
?
Sloppy sketch:
[attachment=0:1x61ry4a]Bead.JPG[/attachment:1x61ry4a]
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MY EDIT: Added clarification about Real-number answer, and then I came back and added it properly. :roll:
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Good one, mmm444bot!
That is one of my favorites . . .
Here's another:
A hiker starts at the center \(\displaystyle A\) of a circular park with radius 17 miles.
He hikes 11 miles directly east to point \(\displaystyle B.\)
Then he hikes directly north until he meets the circumference of the park, point \(\displaystyle C.\)
Finally he hikes directly west to point \(\displaystyle D\), directly north of \(\displaystyle A.\)
Find the distance \(\displaystyle BD.\)
(Time limit: one minute)
That is one of my favorites . . .
Here's another:
A hiker starts at the center \(\displaystyle A\) of a circular park with radius 17 miles.
He hikes 11 miles directly east to point \(\displaystyle B.\)
Then he hikes directly north until he meets the circumference of the park, point \(\displaystyle C.\)
Finally he hikes directly west to point \(\displaystyle D\), directly north of \(\displaystyle A.\)
Find the distance \(\displaystyle BD.\)
(Time limit: one minute)
JuicyBurger
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Actually, you do need to know more information. You need to at least know the radius of the sphere.
With the information given, the sphere could be very, very large and drilled by a very, very large diameter cylinder.
It could also be a very, very small sphere ( > 6 units diameter) with a smaller cylinder drilled out from it.
Could you clarify?
With the information given, the sphere could be very, very large and drilled by a very, very large diameter cylinder.
It could also be a very, very small sphere ( > 6 units diameter) with a smaller cylinder drilled out from it.
Could you clarify?
JuicyBurger
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Actually, I think I may be mistaken ( =D) I'm still working it out...
JuicyBurger
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soroban said:
\(\displaystyle 17 \ miles\)
mmm4444bot
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Soroban, you trickster!
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Reading that puzzle description reminded me of another puzzle that I first saw in Polya's "How to Solve It".
It goes something like:
You begin hiking 5 miles south, after which you turn left and hike 5 miles east, after which you turn left and hike 5 miles north. You're now back to where you began, so what color is the bear?
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Aladdin, are you solving all of these suggestions?
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MY EDIT: Fixed hiking directions. (I must have been standing on my head.)
soroban said:
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Soroban, you trickster!
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Reading that puzzle description reminded me of another puzzle that I first saw in Polya's "How to Solve It".
It goes something like:
You begin hiking 5 miles south, after which you turn left and hike 5 miles east, after which you turn left and hike 5 miles north. You're now back to where you began, so what color is the bear?
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Aladdin, are you solving all of these suggestions?
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MY EDIT: Fixed hiking directions. (I must have been standing on my head.)
mmm4444bot
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JuicyBurger said:
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As the spheres get larger, the volume drilled away to obtain a hole with a 6-unit length increases proportionately. So, it always turns out to be the same remaining volume, regardless of the sphere's initial size.
?[/spoiler:14ehdikp]
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As the spheres get larger, the volume drilled away to obtain a hole with a 6-unit length increases proportionately. So, it always turns out to be the same remaining volume, regardless of the sphere's initial size.
?[/spoiler:14ehdikp]
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Geometry or calculus will lead you to the correct answer. Geometrically, let the radius of the sphere be R. The height of the remaining volume is given as 6. The radius of the cylindrical hole created is then sqrt(R^2 - 9). The height of the spherical end caps at the ends of the cylindrical hole is (R - 3). The remaining volume is then the total volume of the sphere minus the volume of the cylindrical hole minus the volume of the two end caps. The volume of the cylindrical hole is Vh = 6Pi(R^2 - 9). The volume of the end cap is Vc = Pih(3R^2 + h^2)/6. Substituting, expanding, and simplifying, the remaining volume becomes 36Pi, a quantity totally independent of the radius of the sphere. Working it out in general terms with a hole length L, the remaining volume becomes Vr = PiL^3/6.
John W. Cambell Jr., editor of Astounding Science Fiction, probably offered the best explanation of the hole in the sphere problem for the untrained mathematical eye. The problem would never have been created unless it had a unique solution. If it has a unique solution, the volume must be a constant which would hold even when the hole is reduced to zero radius. Therefore the residue must equal the volume of a sphere with a diameter of six inches, namely 36Pi. In other words, the residue is constant regardless of the hole's diameter or the size of the sphere.
John W. Cambell Jr., editor of Astounding Science Fiction, probably offered the best explanation of the hole in the sphere problem for the untrained mathematical eye. The problem would never have been created unless it had a unique solution. If it has a unique solution, the volume must be a constant which would hold even when the hole is reduced to zero radius. Therefore the residue must equal the volume of a sphere with a diameter of six inches, namely 36Pi. In other words, the residue is constant regardless of the hole's diameter or the size of the sphere.
Re:
Hello, mmm4444bot!
Did you know that there is an alternate answer to this riddle?
I'm sure everyone knows this riddle, so I'm not giving anything away, am I?
The standard answer is "White".
The bear is a polar bear because this hiking took place at the North Pole.
However, there are a brizillion* places where this can happen.
Start at a point somewhere near the South Pole.
You hike 5 miles north, then 5 miles west.
And you walk around a circle of laltitude with a circumference of exactly five miles.
Then you hike 5 miles south and arrive at your starting point.
The answer is: There are no bears in Antarctica.
You can start at a point even closer to the South Pole,
so that when you walk 5 miles west,
you walk twice around that circle of latitude.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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How much is a brizillion?
A what?
A brizillion . . . I just heard it on the new:
"Two brizilian soldiers injured."
Hello, mmm4444bot!
Did you know that there is an alternate answer to this riddle?
I'm sure everyone knows this riddle, so I'm not giving anything away, am I?
The standard answer is "White".
The bear is a polar bear because this hiking took place at the North Pole.
However, there are a brizillion* places where this can happen.
Start at a point somewhere near the South Pole.
You hike 5 miles north, then 5 miles west.
And you walk around a circle of laltitude with a circumference of exactly five miles.
Then you hike 5 miles south and arrive at your starting point.
The answer is: There are no bears in Antarctica.
You can start at a point even closer to the South Pole,
so that when you walk 5 miles west,
you walk twice around that circle of latitude.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
*
How much is a brizillion?
A what?
A brizillion . . . I just heard it on the new:
"Two brizilian soldiers injured."
mmm4444bot
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Oops, I gave wrong directions. (I'll fix that.)
Actually, the best hiking directions are sometimes, "Just stay in the car".
[attachment=1:1m0fkt4x]HikingWithBears.JPG[/attachment:1m0fkt4x]
[attachment=0:1m0fkt4x]HikingWithEagles.JPG[/attachment:1m0fkt4x]
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Here's something else, for Soroban and Denis. Saw this in a Bahamas newspaper. :wink:
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soroban said:
Oops, I gave wrong directions. (I'll fix that.)
Actually, the best hiking directions are sometimes, "Just stay in the car".
[attachment=1:1m0fkt4x]HikingWithBears.JPG[/attachment:1m0fkt4x]
[attachment=0:1m0fkt4x]HikingWithEagles.JPG[/attachment:1m0fkt4x]
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Here's something else, for Soroban and Denis. Saw this in a Bahamas newspaper. :wink:
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Code:
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDWESTDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD
DDDDDDDDDDDDDDDDDDDDDDDDDDDD?
Attachments
mmm4444bot
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This looks like too much stuff got squared. (Or, maybe I'm 2^2.)
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?Denis said:
This looks like too much stuff got squared. (Or, maybe I'm 2^2.)
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