Math Puzzle -- Northwest Exposure
July 11th, 2012 at 1:11:40 PM
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You start at any point on the equator and head northwest until you reach the North Pole.
1. What is the distance of your journey? Express your answer as a function of c, where c = the circumference of the earth.
2. How many revolutions around the earth do you make?
1. What is the distance of your journey? Express your answer as a function of c, where c = the circumference of the earth.
2. How many revolutions around the earth do you make?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 11th, 2012 at 1:15:39 PM
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Quote: WizardYou start at any point on the equator and head northwest until you reach the North Pole.
1. What is the distance of your journey? Express your answer as a function of c, where c = the circumference of the earth.
2. How many revolutions around the earth do you make?
I'm not sure I understand the parameters.
If you start on the equator and head off at an angle of 45 degrees, you will complete a circle but never reach the North Pole.
Simplicity is the ultimate sophistication - Leonardo da Vinci
July 11th, 2012 at 1:24:33 PM
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Quote: AyecarumbaI'm not sure I understand the parameters.
If you start on the equator and head off at an angle of 45 degrees, you will complete a circle but never reach the North Pole.
If you always travel NW, you angle relative to the latitude will change.
"Then you can admire the real gambler, who has neither eaten, slept, thought nor lived, he has so smarted under the scourge of his martingale, so suffered on the rack of his desire for a coup at trente-et-quarante" - Honore de Balzac, 1829
July 11th, 2012 at 1:29:18 PM
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I don't have the math skills for this, but based upon Aye's spoiler, I will state this:
1. You are not heading at an angle of 45, but are always heading northwest. Aye's path is a straight line over the surface, forming a circle. The correct path is a spiral heading around the pole and north towards it.
2. You make zero revolutions around the earth, (or maybe that should be 1/4 - from the equator to the pole), but you make several (many?) revolutions around the pole.
2. You make zero revolutions around the earth, (or maybe that should be 1/4 - from the equator to the pole), but you make several (many?) revolutions around the pole.
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July 11th, 2012 at 1:34:47 PM
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Quote: WizardYou start at any point on the equator and head northwest until you reach the North Pole.
1. What is the distance of your journey? Express your answer as a function of c, where c = the circumference of the earth.
2. How many revolutions around the earth do you make?
I'm guessing part 2 is easy.
I think you'll make an infinite number of revolutions around the earth but in a finite amount of time!
July 11th, 2012 at 1:41:14 PM
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My gut response is that it will take exactly 1 revolution
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July 11th, 2012 at 1:49:06 PM
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There will be an infinite number of revolutions around the North pole. I am not sure if they can really be considered "around the Earth", the path will look more like a spiral, converging towards the pole.
As for the total distance, I get c*sqrt(2)/4, I think.
"When two people always agree one of them is unnecessary"
July 11th, 2012 at 2:46:15 PM
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Let me clarify that a "revolution around the earth" is the same as a revolution around the north pole. At least for purposes of this problem.
After incorrectly stating the wrong answer to my last math puzzle, I'm not going to say definitively that my answer is right. That said, I think that weaselman has the correct answer to part 1, and ChesterDog to part 2. If you click the spoiler buttons you will get zero credit, unless to verify your own correct answer.
After incorrectly stating the wrong answer to my last math puzzle, I'm not going to say definitively that my answer is right. That said, I think that weaselman has the correct answer to part 1, and ChesterDog to part 2. If you click the spoiler buttons you will get zero credit, unless to verify your own correct answer.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 11th, 2012 at 2:54:29 PM
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I am not sure that the distance can be calculated, as one is continually heading away from the point that represents the North Pole, and, hence, one will never reach that point.
I would guess that the number of revolutions around the pole would also involve infinity.
I would guess that the number of revolutions around the pole would also involve infinity.
July 11th, 2012 at 3:43:33 PM
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To those who say you would never reach the North Pole, due to an infinite number of spirals, let me ask this problem #2.
There is one ant on each corner of a square table of side length 1 meter. Each ant walks at rate of one meter per minute. Each ant directly crawls towards the ant on his left.
2a. How long does each ant crawl before meeting in the middle of the table?
2b. How many spirals around the center does each ant make?
To avoid confusion, please refer to the original questions as 1a and 1b.
There is one ant on each corner of a square table of side length 1 meter. Each ant walks at rate of one meter per minute. Each ant directly crawls towards the ant on his left.
2a. How long does each ant crawl before meeting in the middle of the table?
2b. How many spirals around the center does each ant make?
To avoid confusion, please refer to the original questions as 1a and 1b.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 11th, 2012 at 4:10:53 PM
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Quote: WizardTo those who say you would never reach the North Pole, due to an infinite number of spirals, let me ask this problem #2.
There is one ant on each side of a square table of side length 1 meter. Each ant walks at rate of one meter per minute. Each ant directly crawls towards the ant on his left.
2a. How long does each ant crawl before meeting in the middle of the table?
2b. How many spirals around the center does each ant make?
To avoid confusion, please refer to the original questions as 1a and 1b.
Here is my response:
The question does not state that the ants are equidistant, nor that they are located in the midpoint of each side of the table. Ant1 and Ant2 could be millimeters apart at one corner of the table, while Ant3 and Ant4 could both be near the opposite corner. In this case, the ant on the right (call him Ant1) in the Ant1/Ant2 pair will overtake Ant2 since he will be coming toward Ant 1 in pursuit of Ant3 (who also overtakes Ant4).
So, they don't necessarily first meet in the middle.
So, they don't necessarily first meet in the middle.
Simplicity is the ultimate sophistication - Leonardo da Vinci
July 11th, 2012 at 4:20:12 PM
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Quote: WizardLet me clarify that a "revolution around the earth" is the same as a revolution around the north pole. At least for purposes of this problem.
After incorrectly stating the wrong answer to my last math puzzle, I'm not going to say definitively that my answer is right. That said, I think that weaselman has the correct answer to part 1, and ChesterDog to part 2. If you click the spoiler buttons you will get zero credit, unless to verify your own correct answer.
Is the "Pole" a point, or a line?
As I understand it, the Earth does, "wobble" as it spins, so there is a good chance you could run into the Pole it if it is a line, and if you are off the surface of the Earth (in a plane or a rocket).
Simplicity is the ultimate sophistication - Leonardo da Vinci
July 11th, 2012 at 4:30:59 PM
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Quote: WizardTo those who say you would never reach the North Pole, due to an infinite number of spirals, let me ask this problem #2.
This problem seems like a similar infinity paradox.
Assuming that all of the ants are identical, they all must follow identical spiral paths. So, the only point at which they could possibly converge would have to be the center of the table. However, each ant is continually moving away from the table's center, chasing another ant (to the left) which is traveling away from it.
Each ant represents a point in a constantly shrinking (and constantly rotating) square, that is always centered around the table's center point, regardless of how infinitesimal the square gets.
So, the ants never reach each other and they never reach the center point on the table.
At least, that's how I see it.
Assuming that all of the ants are identical, they all must follow identical spiral paths. So, the only point at which they could possibly converge would have to be the center of the table. However, each ant is continually moving away from the table's center, chasing another ant (to the left) which is traveling away from it.
Each ant represents a point in a constantly shrinking (and constantly rotating) square, that is always centered around the table's center point, regardless of how infinitesimal the square gets.
So, the ants never reach each other and they never reach the center point on the table.
At least, that's how I see it.
July 11th, 2012 at 4:34:16 PM
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Regarding problem 2, let me clarify that in the initial state there is one ant on each corner.
To those who say the ants will never reach the middle, consider the function of the distance between adjacent ants by time.
To those who say the ants will never reach the middle, consider the function of the distance between adjacent ants by time.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 11th, 2012 at 4:36:03 PM
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Quote: WizardRegarding problem 2, let me clarify that in the initial state there is one ant on each corner. To those who say the ants will never reach the middle, consider the function of the distance between adjacent ants by time.
Are the ants infinitely small -- like a point?
July 11th, 2012 at 4:39:06 PM
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The first problem is an analog one, the second a digital one, with a "bit" being the distance of an ant's step.
A falling knife has no handle.
July 11th, 2012 at 4:42:40 PM
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How west is northwest?
If you're facing directly north, that's north. If you face one degree westward from north, is that northwest? Does that affect anything?
If you're facing directly north, that's north. If you face one degree westward from north, is that northwest? Does that affect anything?
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July 11th, 2012 at 4:46:41 PM
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Quote: tuppAre the ants infinitely small -- like a point?
Yes.
Quote: NareedHow west is northwest?
Assume you have a compass and always head exactly northwest. Meaning the line halfway between north and west.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 11th, 2012 at 4:48:05 PM
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Quote: NareedIf you're facing directly north, that's north. If you face one degree westward from north, is that northwest? Does that affect anything?
If one is continually traveling away from a point, one can never reach that point.
July 11th, 2012 at 4:57:23 PM
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Quote: tuppIf one is continually traveling away from a point, one can never reach that point.
Sort of like how an infinite string of nines in "0.999999999..." will never reach "1"?
Simplicity is the ultimate sophistication - Leonardo da Vinci
July 11th, 2012 at 5:04:07 PM
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Quote: NareedHow west is northwest?
If you're facing directly north, that's north. If you face one degree westward from north, is that northwest? Does that affect anything?
Northwest (N45W) is exactly 315 degrees clockwise from the North. The direction one degree westward from North is properly referred to as "359 degrees", "minus one degree", or N1W.
Why they measure it clockwise and in degrees, while about any other angle is counterclockwise and in radians, is one of life's little mysteries.
"When two people always agree one of them is unnecessary"
July 11th, 2012 at 5:04:44 PM
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I like these problems, but I have a problem with their unrealistic application to real life.
At first, for the distance you would travel, my conceptual answer would've been the circumference of the earth (assuming a perfect sphere). Then someone said infinite, and it clicked that you shouldn't ever reach the North Pole. My answer would assume that at some point you're "close enough" and would just stop.
I would imagine, that the distance is a convergence problem. The actual distance should converge to a finite distance. At least that's what I think of conceptually.
My biggest problem is, if you're looking at a person walking, or ants walking on a table, they take up physical space, and the physical constraints prevent them to continue walking, since eventually they would need to make infinitesimally small movements. I realize that's not really the point, but it still bothers me.
I would imagine, that the distance is a convergence problem. The actual distance should converge to a finite distance. At least that's what I think of conceptually.
My biggest problem is, if you're looking at a person walking, or ants walking on a table, they take up physical space, and the physical constraints prevent them to continue walking, since eventually they would need to make infinitesimally small movements. I realize that's not really the point, but it still bothers me.
July 11th, 2012 at 5:21:11 PM
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Quote:Are the ants infinitely small -- like a point?
Quote:Yes.
Okay. It would seem that infinitely small points are always traveling away from the center, chasing another point which is always fleeing and which is also traveling away from the center. Such is always the case, no matter how small this scenario shrinks and regardless of speed and time.
I would guess that the spiral path that the points take would always be the same, regardless of their speed.
The points can never overtake each other nor can they reach the table's center. So, to me, it seems that speed and distance (paradoxically) would not affect this outcome.
I would guess that the spiral path that the points take would always be the same, regardless of their speed.
The points can never overtake each other nor can they reach the table's center. So, to me, it seems that speed and distance (paradoxically) would not affect this outcome.
July 11th, 2012 at 5:43:18 PM
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Quote: tuppIf one is continually traveling away from a point, one can never reach that point.
I would argue that the ants are traveling in a spiral drawing closer to the center.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 11th, 2012 at 7:32:00 PM
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Can't get this problem out of my head. I hope this makes sense since I'm not really giving a full answer, but still using Calculus concepts.
For some reason intuition tells me the distance you would travel would be ~c. Of course, since you can never actually reach the North Pole, a more correct answer would be the distance converges to C. Here's my reasoning:
Let's draw a longitude from the North Pole to our starting point on the Equator. Each time our path crosses that longitude let's say we have travelled one "revolution". Our revolutions will be infinite. However, each revolution will be a shorter distance. I can't find my Calc book, so I'm not sure of the proper way to state this, so I'll use limits.
NOTE: The next part has a lot of guess work.
Let's say for the first revolution we have travelled 1/3 of C. Our next revolution we travel will be 1/3 of 2/3 of C. The next revolution will be 1/3 of the remaining C we have not travelled, and so on. So, if we define a fuction F(x), where x is the number of revolutions, our lim as x approaches infinity would = C. Not exactly sure what the fuction would look like. I feel like what I want is floating up in my head somewhere, but I can't quite access it. F(x) = x/3?? That doesn't seem right. But does anyone see my point.
I'm probably way off, but that's how I see it. It's been about six years since I've taken Calculus and that's what type of problem I think this is.
Let's draw a longitude from the North Pole to our starting point on the Equator. Each time our path crosses that longitude let's say we have travelled one "revolution". Our revolutions will be infinite. However, each revolution will be a shorter distance. I can't find my Calc book, so I'm not sure of the proper way to state this, so I'll use limits.
NOTE: The next part has a lot of guess work.
Let's say for the first revolution we have travelled 1/3 of C. Our next revolution we travel will be 1/3 of 2/3 of C. The next revolution will be 1/3 of the remaining C we have not travelled, and so on. So, if we define a fuction F(x), where x is the number of revolutions, our lim as x approaches infinity would = C. Not exactly sure what the fuction would look like. I feel like what I want is floating up in my head somewhere, but I can't quite access it. F(x) = x/3?? That doesn't seem right. But does anyone see my point.
I'm probably way off, but that's how I see it. It's been about six years since I've taken Calculus and that's what type of problem I think this is.
July 11th, 2012 at 8:46:56 PM
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" like these problems, but I have a problem with their unrealistic application to real life. "
I wish you had told me that before I wasted 3 hours trying to train these dumb ants.
I wish you had told me that before I wasted 3 hours trying to train these dumb ants.
July 12th, 2012 at 7:37:46 AM
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To the North Pole question:
I really thought this would be more difficult, but it came clear to me this morning.
I really thought this would be more difficult, but it came clear to me this morning.
If you walk directly north, you will need to travel c/4. If you walk northwest, you will be constantly walking at a 45 degree angle. This is the part that confused me the most, but as soon as I resolved that with a few sketches, the answer became clear. To find the answer, draw a right triangle. The hypotenuse is the distance actaully travelled and one of the legs is c/4. Because you are walking at a 45 degree angle, both legs are c/4, but that doesn't really matter.
At this point, I knew that the hypotenuse is the leg*sqrt(2), but you can use a little trig as well, by calculating the sin of the angle. sin(45) = (c/4)/distance travelled, so distance travelled = (c/4)/sin(45). sin(45)=sqrt(2)/2, and you can take it from there.
At this point, I knew that the hypotenuse is the leg*sqrt(2), but you can use a little trig as well, by calculating the sin of the angle. sin(45) = (c/4)/distance travelled, so distance travelled = (c/4)/sin(45). sin(45)=sqrt(2)/2, and you can take it from there.
I heart Crystal Math.
July 12th, 2012 at 8:12:37 AM
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Quote: CrystalMathTo the North Pole question:
I really thought this would be more difficult, but it came clear to me this morning.
I found the answer by straightforward integrating first, and then realized that it is the same as it would be if one traveled on a plane, and found that fact pretty fascinating :)
But I am still having a hard time convincing myself that your approach is valid - what is your reason to believe that planar trigonometry can be applied to the sphere in this case?
I was thinking about using a conformal map, like mercator projection, on which the path (loxodrome) would indeed look like a hypotenuse of a right triangle ... but that actually makes it worse, because the distances aren't preserved in general. So, why are they in this case? Coincidence?
But I am still having a hard time convincing myself that your approach is valid - what is your reason to believe that planar trigonometry can be applied to the sphere in this case?
I was thinking about using a conformal map, like mercator projection, on which the path (loxodrome) would indeed look like a hypotenuse of a right triangle ... but that actually makes it worse, because the distances aren't preserved in general. So, why are they in this case? Coincidence?
"When two people always agree one of them is unnecessary"
July 12th, 2012 at 8:24:55 AM
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Quote: weaselmanI found the answer by straightforward integrating first, and then realized that it is the same as it would be if one traveled on a plane, and found that fact pretty fascinating :)
But I am still having a hard time convincing myself that your approach is valid - what is your reason to believe that planar trigonometry can be applied to the sphere in this case?
I was thinking about using a conformal map, like mercator projection, on which the path (loxodrome) would indeed look like a hypotenuse of a right triangle ... but that actually makes it worse, because the distances aren't preserved in general. So, why are they in this case? Coincidence?
If you consider an infinitesimal step, the curvature of the planet has zero impact. If your step is length ds, and your angle is 45 degrees, then you will get sqrt(2)/2*ds closer to the pole with that step.
I heart Crystal Math.
July 12th, 2012 at 8:32:00 AM
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For the ant problem:
2a: The ants will travel 1 meter. Again, a bit difficult to wrap my brain around. Let's say you are the first ant. The second ant, that you are trailing, is moving orthogonal to you, so he is not moving closer to you or further from you, only you are moving closer to him. Because each step you take is moving you directly toward the ant you are trailing, then you will need to cover the initial distance between you, which is 1 meter.
2b: Infinite spirals around the center of the table.
2b: Infinite spirals around the center of the table.
I heart Crystal Math.
July 12th, 2012 at 8:58:31 AM
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Quote: WizardYou start at any point on the equator and head northwest until you reach the North Pole.
1. What is the distance of your journey? Express your answer as a function of c, where c = the circumference of the earth.
2. How many revolutions around the earth do you make?
I might say that this problem only becomes extremely abstract after a half a turn of the earth. So as an addendum problem what if you go through 1/2 revolution (i.e 180 degree of longitude) following a rhumb line with a constant 45 degree of bearing. What is the distance traveled as a function of c? As a percent of the answer to the original question (i.e. reaching the north pole)? What degree of latitude will you have reached?
The spiraling is a function of any degree of bearing (not just 45 degrees). The 45 on this picture is not the degree of bearing of the rhumb line, but the angle of orientation of the globe for the purposes of visualization.
July 12th, 2012 at 9:01:00 AM
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Quote: CrystalMath
If you consider an infinitesimal step, the curvature of the planet has zero impact. If your step is length ds, and your angle is 45 degrees, then you will get sqrt(2)/2*ds closer to the pole with that step.
For infinitesimal step, yes.
But how do you get from infinitesimal to finite without integration - that's the question.
"When two people always agree one of them is unnecessary"
July 12th, 2012 at 9:26:56 AM
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Quote: weaselmanQuote: CrystalMath
If you consider an infinitesimal step, the curvature of the planet has zero impact. If your step is length ds, and your angle is 45 degrees, then you will get sqrt(2)/2*ds closer to the pole with that step.
For infinitesimal step, yes.
But how do you get from infinitesimal to finite without integration - that's the question.
This one just makes intuitive sense to me. You must travel c/4 north, which is a curved path placed onto plane. Each step is at 45 degrees from north.
Could you post your work? My first thought was that it would require integration, but frankly, I didn't know where to start since I haven't used Calculus in 15 years.
Could you post your work? My first thought was that it would require integration, but frankly, I didn't know where to start since I haven't used Calculus in 15 years.
I heart Crystal Math.
July 12th, 2012 at 9:46:34 AM
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Quote: CrystalMath
Could you post your work? My first thought was that it would require integration, but frankly, I didn't know where to start since I haven't used Calculus in 15 years.[/spoiler]
It's pretty straightforward really ...
ds = sqrt(dx^2 + dy^2)
dy = R*d(phi)
dx = dy * tan(a) = dy
where phi is latitude, and a is the azimuth (45 degrees). So:
ds = R*sqrt(2*d(phi)^2) = R*d(phi)*sqrt(2)
Integrate that from 0 to pi/2 to get pi*R*sqrt(2)/2
"When two people always agree one of them is unnecessary"
July 12th, 2012 at 9:53:07 AM
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Quote: weaselmanQuote: CrystalMath
Could you post your work? My first thought was that it would require integration, but frankly, I didn't know where to start since I haven't used Calculus in 15 years.[/spoiler]
It's pretty straightforward really ...
ds = sqrt(dx^2 + dy^2)
dy = R*d(phi)
dx = dy * tan(a) = dy
where phi is latitude, and a is the azimuth (45 degrees). So:
ds = R*sqrt(2*d(phi)^2) = R*d(phi)*sqrt(2)
Integrate that from 0 to pi/2 to get pi*R*sqrt(2)/2
Cool. Makes perfect sense now. Crazy simple integration, I didn't need a calc book for that one. Thanks.
I heart Crystal Math.
July 12th, 2012 at 11:30:17 AM
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I will be unable to post my answer today. One on my ants died.
July 12th, 2012 at 12:35:01 PM
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oops. never mind
"When two people always agree one of them is unnecessary"
July 16th, 2012 at 4:44:01 AM
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This was an interesting problem. The way I saw it was that while the path was on a sphere at any given moment you can ignore that effect, as if the earth were flat, as CrystalMath explained. That would simply increase the distance by sqrt(2). So my answer is c*sqrt(2)/4 and infinite loops.
At the risk of waking a sleeping dog, I think that if you can grasp how infinite loops can be made in a finite amount of time you can grasp why .999... = 1.
At the risk of waking a sleeping dog, I think that if you can grasp how infinite loops can be made in a finite amount of time you can grasp why .999... = 1.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 16th, 2012 at 8:44:57 AM
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Quote: WizardThis was an interesting problem. The way I saw it was that while the path was on a sphere at any given moment you can ignore that effect, as if the earth were flat, as CrystalMath explained. That would simply increase the distance by sqrt(2). So my answer is c*sqrt(2)/4 and infinite loops.
At the risk of waking a sleeping dog, I think that if you can grasp how infinite loops can be made in a finite amount of time you can grasp why .999... = 1.
It is a bit difficult, still. I didn't think much of it until the ant problem. The ants will travel
1 meter, which will take 1 minute
I heart Crystal Math.
July 16th, 2012 at 9:21:43 AM
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I agree with CM's answer to the ant problem above. Indeed, I find it hard to get my mind around the fact that infinite loops can be made in a finite amount of time, but they can. When it comes to infinity, there are lots of such paradoxes.
Time for a new problem.
Time for a new problem.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 16th, 2012 at 9:31:38 AM
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Quote: Wizard... I find it hard to get my mind around the fact that infinite loops can be made in a finite amount of time, but they can. When it comes to infinity, there are lots of such paradoxes.
I suggest that you think of them as an infinite number of loops, almost all of which have almost zero length each. The total length is such that it may be traversed in a modest amount of time. You just might get quite dizzy in the process.
July 16th, 2012 at 9:38:33 AM
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Quote: DocI suggest that you think of them as an infinite number of loops, almost all of which have almost zero length each. The total length is such that it may be traversed in a modest amount of time. You just might get quite dizzy in the process.
I do think of it that way.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
July 20th, 2012 at 6:30:28 PM
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Sorry if this has already been said but here was my simple logic.Quote: Wizard...my answer is c*sqrt(2)/4 and infinite loops...
Assume you only move a small amount NorthWest each time. Consider a very small right-angled triangle, (OB) 1 unit North, (OA) 1 unit West, then (AB) = sqrt(2) units. For small values, you travel along A to B - so for each sqrt(2) travelled get 1 nearer to the pole. Since the straight line distance from equator to pole is c/4, the distance travelled is c*sqrt(2)/4.
July 20th, 2012 at 7:56:06 PM
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Stop It!!Quote: AyecarumbaSort of like how an infinite string of nines in "0.999999999..." will never reach "1"?
Someday, joor goin' to see the name of Googie Gomez in lights and joor goin' to say to joorself, "Was that her?" and then joor goin' to answer to joorself, "That was her!" But you know somethin' mister? I was always her yuss nobody knows it! - Googie Gomez
