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Do you mean this problem is taken from some book and 400 km s the answer when time and distance are given ?
If so, why does he ask this question again ?
I am sorry I just think because this thread title is Questions and Answers, there must be someone who couldnot do his/her homework joinin to make questions and waiting for help from some others...
No, the answer is 55,... or not!
I am not sure whether point C is supposed to be a space coordinate, a time coordinate, or a space-time coordinate.
C as a space coordinate:
* The problem states that "As a whole he went really uneven in both directions". I assume this means that his paths were random. C cannot exist a space coordinate with this interpretation.
** This means that his two paths do not have to cross (when viewed as space paths).
** Since his two paths do not have to cross, then the only points in common at the end-points of his travel, the two monasteries (points A and B) and there cannot in general be a midpoint C common to both paths.
* If "As a whole he went really uneven in both directions" means that the monk followed the same path, but just traveled it at different speeds, (and we consider space to be a static Euclidean space) then the answer would be yes, point C can exist as a space coordinate because time is now irrelevant (we are only concerned with space coordinates.)
** Choose a point C that is equidistant from A and B for the first trip so that distance(AC) = distance(CB).
** Since the space is static and Euclidean, distance(XY) = distance(YX) for all time.
** So we may rewrite the equation distance(AC) = distance(CB) as distance(CA) = distance(BC).
** Using the math identity (I forgot the name for this) that if X=Y then Y=X, we can rewrite distance(CA) = distance(BC) as distance(BC) = distance(CA).
** But the equation distance(BC) = distance(CA) describes the return trip. So yes, point C exists as the midpoint in space coordinates assuming (a) the monk followed the same space-path and (b) we are assuming a static Euclidean space.
* I suppose there exists the possibility that we are dealing with static non-Euclidean space. I am not sure that in general the previous "proof" (and I use that term lightly) would be true. Then we could also be talking about a non-static space. But really, this means that the supposed path itself changes and my first "proof" would be sufficient here as well to state that point C cannot exist,
C as a time coordinate:
* Assuming that the monk is not infinately fast at traveling and working, the second travel happened after his first travel. C cannot exist as a time coordinate with this setup.
** This means we have t(A1) < t(B1) < t(B2) < t(A2).
** There is no solution where a fixed t(C) exist so that t(A1) < t(C) < t(B1) < t(B2) < t(C) < t(A2).
** So t(C) cannot exist as a mid-point for both time paths.
* If the monk is infinately fast at traveling and working, then t(A1) = t(C) = t(B1) = t(B2) = t(C) = t(A2), and in this case C does exist as a time coordinate because all time coordinates are the same.
C cannot exist as a space-time coordinate in general:
* C can only exist as a space-time coordinate if the monk is infinately fast at traveling and working and travels the same path both ways.
** That is the only solution that satisfies C as time coordinate.
** If everything happend as the same time, then we only need to consider the space at one point in time. This removes all concern about whether the space is static or not.
** Then the only solution possible from a space-viewpoint is that the monk follow the same path (see first agument for C as a space coordinate).
** But if the monk traveled infinately fast and traveled the same space path, then he really did not travel "uneven in both directions."
** So we have generated a contridiction and proved that C cannot exist as a space-time coordinate in general.
Kevin, I assumed that when SevenStar says:
"point (C) that is positioned in time so that AC (in time) is equal BC (in time) in the monk's walk."
he talks about a time of walk relative to the beginning of each travel. It means that when he leaves A, the time is set to zero, for his first trip, and when he leaves B on his way back, the time is set to zero for his second trip.
In this case, if he takes the same path on both ways, there will be a point where t(AC) = t(BC).
Now, as the monk waks unevenly, I don't think it is possible to know exactly where that point is. The best idea would says that it is somewhere in the section of his first trip between his start from A and the point he has reached after walking the same length of time that he has walked on his way back.
Well then is C a point in space, or a point in time (where time is reset to zero)? If C is a point in space, there is nothing to say that the same space-point C will equally divide the monk's path in time.Quote:
Originally posted by Elrond
Kevin, I assumed that when SevenStar says:
"point (C) that is positioned in time so that AC (in time) is equal BC (in time) in the monk's walk."
he talks about a time of walk relative to the beginning of each travel. It means that when he leaves A, the time is set to zero, for his first trip, and when he leaves B on his way back, the time is set to zero for his second trip.
In any case I made assumptions, you made assumptions. If people are making too many assumptions, then the question needs clarification. Seventh Star, is C supposed to be a space point or a time point (or a space-time point)? Is time reset to zero at the beginning of each trip? What exactly do you mean by As a whole he went really uneven in both directions"? Did we walk unevenly in speed, path, or both?
Right, the answer needs clarification. And what is the expected answer? The exact location of the point C? Or the best possible estimation?
Rolle's theorem (a version of the mean value theorem)
How does Rolle's theorem apply here?
What is Rolle's theorem?
If f(x) is continuous on [a,b] and differntiable on (a,b) and f(a)=f(b), then there is some point c that is part of the interval (a,b) where f'(c)=0.
Well there cannot be any possible estimation becaouse the monk was going very unsteady...Quote:
Originally posted by Elrond
Right, the answer needs clarification. And what is the expected answer? The exact location of the point C? Or the best possible estimation?
Sorry for the second questioen it wasn't correctly put :rolleyes:
Anyway the expected answer was:
"If you have two monks that are the same and they both start going simultaniously the first from A to B and the second from B to A, they will meet in point C"
I see now. The question now makes more sense now that I know the answer. Interesting question. A few assumptions did need to be made (like that the monk followed the same path in both of his walks). What's thenext one?
Who knows? :D I don't!Quote:
Originally posted by KevinHall
What's thenext one?
If you have a question, go ahead!
Rolle's theorem is equivalent to the intermediate value theorem (equivalent in the mathematical sense that if one assumes the validity of one, all valid conclusions of the axiom set are conclusions to if one assumes the other). They both codify the topological / analytic result that if a continuous function (like the assumption that the monk cannot teleport) takes on values f(a) and f(b) (with f(a) < f(b)), then for any value f(a) < c < f(b) there exist a value d between a and b such that f(d) = c. It is the mathematical theorem required to prove SeventhStar's "If you have two monks that are the same and they both start going simultaniously the first from A to B and the second from B to A, they will meet in point C" (which in my opinion is only a nonrigorous restatement of the problem ;)).
Just being cryptic to increase (decrease?) my coolness factor...
Galathaea.... Your avatar is a beauty :)