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where are their destinations? is that assumed to be uniform along the island?
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are we talking about people entering the island or leaving the island as well... or is it half/half
if it's just entering, x = L
if you're coming from the north, you use the north entrance
if you're coming from the south, you use the south entrance
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If each bridge can take any volume of traffic, then I supoose + Show Spoiler + seems like an obvious choice.
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On September 13 2009 10:37 deconduo wrote:If each bridge can take any volume of traffic, then I supoose + Show Spoiler + seems like an obvious choice.
yep! though i suck at math, this would mean everyone has at most 1/3rd L distance to the nearest route to the mainland
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Well the farthest points are the end of the landmass and the mid-section of the bridge and landmass entry. So my first answer would be x=2L/3
But then I figured that there are two ways to get to the midsection and 1 to get to the end of the mass. So surely x>2L/3.
The difficulty of this problem is interpreting mathematically what he means by "accesibility". I will use physics consider it being the smallest time it would take for people to fill the bridge. Suppose the crowd has velocity v and enters the bridge and the entrance at the same time.
For the people coming from the North:
d_1=L-vt
From the bridge going up:
d_2=(L-x)+vt
From the bridge going down:
d_3=(L-x)-vt
so d_1+d_2+d_3=L . Therefore:
L=3L-2x-vt
vt=2L-2x
And we have d_1=d_2:
L-vt=(L-x)+vt
2vt=x
vt=x/2
Substituting:
x/2=2L-2x
5x/2=2L
x=4L/5
Didn't revise it, hope it is correct.
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On September 13 2009 10:40 nttea wrote:yep! though i suck at math, this would mean everyone has at most 1/3rd L distance to the nearest route to the mainland
Yep, though only thing that could affect it is volume of traffic. If each bridge can only handle a certain amount its different, because in this case bridge 2 takes twice as much as bridge 1.
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On September 13 2009 10:40 nttea wrote:yep! though i suck at math, this would mean everyone has at most 1/3rd L distance to the nearest route to the mainland
the thing is, the question is asking to make the island evenly accessible to the mainlanders not the other way around. in which case azndsh's answer is correct as that eliminates any possibility of having to retrace your steps, plus we know that the people come off the map so we don't have to worry about those that spawn between both bridges
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United States24751 Posts
Ok just got done with something and now I'm actually thinking about this.
On September 13 2009 11:13 JeeJee wrote:Show nested quote +On September 13 2009 10:40 nttea wrote:On September 13 2009 10:37 deconduo wrote:If each bridge can take any volume of traffic, then I supoose + Show Spoiler + seems like an obvious choice. yep! though i suck at math, this would mean everyone has at most 1/3rd L distance to the nearest route to the mainland the thing is, the question is asking to make the island evenly accessible to the mainlanders not the other way around. in which case azndsh's answer is correct as that eliminates any possibility of having to retrace your steps, plus we know that the people come off the map so we don't have to worry about those that spawn between both bridges
I think accessibility for the mainlanders works out to the same thing as accessibility for those living on the smaller land mass.
Although you are right that (I think) the key here is that people can 'spawn' from above or below the map. One thing I want to think about is what proportion of people are coming from the extreme North/South, and what proportion of people are coming from due west of the bridge (south of the northernmost tip of the landmass and north of the southernmost tip). If the percentage of people coming from due west is significant, then the location of the bridge depends on the relative amounts. If the percentage is negligible (0), then x=L makes sense.
When I was creating the problem, I was thinking more along the lines of the case where x!=L. As such, perhaps I should think about redefining the problem such that people can only spawn above/below certain latitudes.
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u realize that simplifies to x = L right?
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Sorry if I'm not understanding this correctly, but what do you mean by "accessibility" (how do you measure it?) And if it's what I think it is, how is this different from a center of mass problem?
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On September 13 2009 11:28 micronesia wrote:Ok just got done with something and now I'm actually thinking about this. Show nested quote +On September 13 2009 11:13 JeeJee wrote:On September 13 2009 10:40 nttea wrote:On September 13 2009 10:37 deconduo wrote:If each bridge can take any volume of traffic, then I supoose + Show Spoiler + seems like an obvious choice. yep! though i suck at math, this would mean everyone has at most 1/3rd L distance to the nearest route to the mainland the thing is, the question is asking to make the island evenly accessible to the mainlanders not the other way around. in which case azndsh's answer is correct as that eliminates any possibility of having to retrace your steps, plus we know that the people come off the map so we don't have to worry about those that spawn between both bridges I think accessibility for the mainlanders works out to the same thing as accessibility for those living on the smaller land mass.
Ah true, I was thinking asymmetrically (where mainlanders can only appear north or south of the bridges as stated, but islanders' destination could be somewhere in between both bridges as well as north/south if that makes any sense at all..)
Either way, as the problem is currently phrased, it's trivialized to x=L. Now if we split the spawn points equally between North, South and West, I believe it becomes a little more complicated..
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United States24751 Posts
On September 13 2009 12:22 Saracen wrote:
Sorry if I'm not understanding this correctly, but what do you mean by "accessibility" (how do you measure it?) And if it's what I think it is, how is this different from a center of mass problem?
Intentionally I'm being a little bit vague about what I mean. The way I define this determines what the answer is. I think I sort of took a stance when I said that cars come from both the extreme North and extreme South.
If we then assume that cars can spawn west of the land mass anywhere between latitudes equivalent to the northernmost and southernmost tips of the land mass, then we've got ourselves a calculation to do as the answer is most likely not trivially x=L.
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