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A difficult math problem...[SOLVED] (pg. 3)
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| Floorfiller |
ok we can do this people hehehe...we're all smart...
there are gonna be two rates at which the rubberband grows in relation to the ant.
1. the section of the rubberband at which he has already travelled
and
2. the section of the rubberband which is in front of him...
at t=1sec, the ant is at .001m and the band at 2m
ok i don't know the math people...you have to help me hehehe, but a portion of that new 1m growth is gonna be within the .001m the ant has already travelled...we need an equation to represent that...
.001(n)/(n+1)= rubberband behind ant :conf:
so then the portion of the rubberband in front of the ant would be that minus n+1 or
[(n+1)-{.001(n)/(n+1)}]= rubberband infront of ant :conf:
ok i don't know what the hell to do from here and that's probably not even right, but i tried hehehe
someone run my equations please and tell me if the cross? |
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| Noisician |
we assume that the ant gets to travel a whole millimeter right before the band stretches by an additional meter. then during the first second, the ant travels 1/1000 of the distance; during the second second it covers 1/2000 of the distance... during the n-th second it covers 1/(n*1000) of the distance. so at the end of the n-th second, the ant has traveled
1/1000 + 1/2000 + 1/3000 + ... + 1/(n*1000) of its way to the end.
after rewriting the series above as
(1/1 + 1/2 +1/3 + ... + 1/n) * 1/1000
we can tell at once that the series diverges because the harmonic series that's in it diverges as well (hopefully no proof necessary?)
so now we know that the ant does in fact reach the end of the rubberband. the only question remans is as follows:
how many terms does it take the harmonic series to add up to 1000?
well, from our math analysis course that we all have taken, we know that the answer is 1.1x10434
it takes this many seconds for the ant to reach the end of the band.
so the answer is yes but it will take the ant approximately 10426 years to get there. |
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| NY1004 |
| quote: | Originally posted by Noisician
we assume that the ant gets to travel a whole millimeter right before the band stretches by an additional meter. then during the first second, the ant travels 1/1000 of the distance; during the second second it covers 1/2000 of the distance... during the n-th second it covers 1/(n*1000) of the distance. so at the end of the n-th second, the ant has traveled
1/1000 + 1/2000 + 1/3000 + ... + 1/(n*1000) of its way to the end.
after rewriting the series above as
(1/1 + 1/2 +1/3 + ... + 1/n) * 1/1000
we can tell at once that the series diverges because the harmonic series that's in it diverges as well (hopefully no proof necessary?)
so now we know that the ant does in fact reach the end of the rubberband. the only question remans is as follows:
how many terms does it take the harmonic series to add to up 1000?
well, from our math analysis course that we all have taken, we know that the answer is 1.1x10434
it takes this many seconds for the ant to reach the end of the band.
so the answer is yes but it will take the ant approximately 10426 years to get there. |
Damn that was pretty sweet man!! I used to fumble over problems like these in my analysis class. Definitely not my forte. Cheers! |
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| Floorfiller |
| quote: | Originally posted by Noisician
we assume that the ant gets to travel a whole millimeter right before the band stretches by an additional meter. then during the first second, the ant travels 1/1000 of the distance; during the second second it covers 1/2000 of the distance... during the n-th second it covers 1/(n*1000) of the distance. so at the end of the n-th second, the ant has traveled
1/1000 + 1/2000 + 1/3000 + ... + 1/(n*1000) of its way to the end.
after rewriting the series above as
(1/1 + 1/2 +1/3 + ... + 1/n) * 1/1000
we can tell at once that the series diverges because the harmonic series that's in it diverges as well (hopefully no proof necessary?)
so now we know that the ant does in fact reach the end of the rubberband. the only question remans is as follows:
how many terms does it take the harmonic series to add up to 1000?
well, from our math analysis course that we all have taken, we know that the answer is 1.1x10434
it takes this many seconds for the ant to reach the end of the band.
so the answer is yes but it will take the ant approximately 10426 years to get there. |
*bows down in homage*
you're the man once again hehehe |
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| Lephaid |
| quote: | Originally posted by Noisician
we assume that the ant gets to travel a whole millimeter right before the band stretches by an additional meter. then during the first second, the ant travels 1/1000 of the distance; during the second second it covers 1/2000 of the distance... during the n-th second it covers 1/(n*1000) of the distance. so at the end of the n-th second, the ant has traveled
1/1000 + 1/2000 + 1/3000 + ... + 1/(n*1000) of its way to the end.
after rewriting the series above as
(1/1 + 1/2 +1/3 + ... + 1/n) * 1/1000
we can tell at once that the series diverges because the harmonic series that's in it diverges as well (hopefully no proof necessary?)
so now we know that the ant does in fact reach the end of the rubberband. the only question remans is as follows:
how many terms does it take the harmonic series to add up to 1000?
well, from our math analysis course that we all have taken, we know that the answer is 1.1x10434
it takes this many seconds for the ant to reach the end of the band.
so the answer is yes but it will take the ant approximately 10426 years to get there. |
yay! thanks. :) |
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| Floorfiller |
noisician...is there any math problem you don't know? :p..
how can i be as smart as you? hehe |
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| Noisician |
| quote: | Originally posted by Floorfiller
noisician...is there any math problem you don't know? :p.. |
oh yes, plenty of them. more than you could ever think of. |
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| physe |
| quote: | Originally posted by Noisician
we assume that the ant gets to travel a whole millimeter right before the band stretches by an additional meter. then during the first second, the ant travels 1/1000 of the distance; during the second second it covers 1/2000 of the distance... during the n-th second it covers 1/(n*1000) of the distance. so at the end of the n-th second, the ant has traveled
1/1000 + 1/2000 + 1/3000 + ... + 1/(n*1000) of its way to the end.
after rewriting the series above as
(1/1 + 1/2 +1/3 + ... + 1/n) * 1/1000
we can tell at once that the series diverges because the harmonic series that's in it diverges as well (hopefully no proof necessary?)
so now we know that the ant does in fact reach the end of the rubberband. the only question remans is as follows:
how many terms does it take the harmonic series to add up to 1000?
well, from our math analysis course that we all have taken, we know that the answer is 1.1x10434
it takes this many seconds for the ant to reach the end of the band.
so the answer is yes but it will take the ant approximately 10426 years to get there. |
Yep, I agree. Although my original post doesn't agree with you, I understand your solution and agree with you now. =)
So no matter how slow the ant goes, it will get there eventually as long as it's moving, right? |
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| Noisician |
| quote: | Originally posted by physe
So no matter how slow the ant goes, it will get there eventually as long as it's moving, right? |
as long as we get the harmonic series or any other series that diverges as a result then yes. and that's the beauty of it. you can think of the harmonic series this way:
suppose we are to fill a bathtub with water. if we take a whole cup of water and put it in the tub and then half a cup, a third, a fourth, a fifth and so on, the bathtub will eventually overflow.
or better yet, suppose we are to fill an empty ocean with water. if we take a whole cup of water and put it in the ocean and then half a cup, a third, a fourth, a fifth and so on, the ocean will eventually overflow. |
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| physe |
| Yeah, I have learnt about the harmonic series and that it diverges, I just didn't think of using it to solve the problem. I also understans that the size of the container doesn't matter as long as the series diverges. |
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| djdawn |
| quote: | Originally posted by Noisician
[color=white]as long as we get the harmonic series or any other series that diverges as a result then yes. and that's the beauty of it. you can think of the harmonic series this way:
suppose we are to fill a bathtub with water. if we take a whole cup of water and put it in the tub and then half a cup, a third, a fourth, a fifth and so on, the bathtub will eventually overflow.
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first, thanks for solving the problem.
now for your example, this only works if you don't take into consideration that at some point, you will only add one water-atom to the cup, and after that you would have to split atoms, right?:conf: |
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| Noisician |
| quote: | Originally posted by djdawn
first, thanks for solving the problem.
now for your example, this only works if you don't take into consideration that at some point, you will only add one water-atom to the cup, and after that you would have to split atoms, right?:conf: |
correct. the only point of that example was to show that we can get an infiinitely large number by adding together infinitely many of ever-decreasing ones. otherwise, you would also have to consider the fact that filling an ocean with water using only a regular cup would take hundreds of millions of years which is virtually impossible. |
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