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Math Problem: Placing a Bridge

Blogs > micronesia
Post a Reply
micronesia
Profile Blog Joined July 2006
United States24670 Posts
Last Edited: 2009-09-13 01:27:03
September 13 2009 01:25 GMT
#1
I just thought this up and it might be trivial or difficult... not sure.

Consider the following picture:

+ Show Spoiler +
[image loading]

I made it in mspaint lol


A land mass that sticks out into the ocean was originally accessible only by a small road at the northern end. A bridge (red) is going to be constructed somewhere along the length of the land mass in order to increase accessibility to the land mass as much as possible. If the length of the land mass is L (thickness of access road negligible), express x, the distance of the bridge from the topmost part of the land mass, in terms of L.

0<x<=L

edit: Assume traffic (or w/e) comes equally from off the map in Both the Northen and Southern directions equally.

*****
ModeratorThere are animal crackers for people and there are people crackers for animals.
azndsh
Profile Blog Joined August 2006
United States4447 Posts
September 13 2009 01:34 GMT
#2
where are their destinations? is that assumed to be uniform along the island?
azndsh
Profile Blog Joined August 2006
United States4447 Posts
September 13 2009 01:36 GMT
#3
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
deconduo
Profile Blog Joined January 2008
Ireland4122 Posts
September 13 2009 01:37 GMT
#4
If each bridge can take any volume of traffic, then I supoose + Show Spoiler +
X=(2/3)L
seems like an obvious choice.
nttea
Profile Blog Joined July 2008
Sweden4353 Posts
September 13 2009 01:40 GMT
#5
On September 13 2009 10:37 deconduo wrote:
If each bridge can take any volume of traffic, then I supoose + Show Spoiler +
X=(2/3)L
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
Z-BosoN
Profile Blog Joined May 2007
Brazil2590 Posts
September 13 2009 01:44 GMT
#6
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.




deconduo
Profile Blog Joined January 2008
Ireland4122 Posts
Last Edited: 2009-09-13 01:47:41
September 13 2009 01:46 GMT
#7
On September 13 2009 10:40 nttea wrote:
Show nested quote +
On September 13 2009 10:37 deconduo wrote:
If each bridge can take any volume of traffic, then I supoose + Show Spoiler +
X=(2/3)L
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


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.
JeeJee
Profile Blog Joined July 2003
Canada5652 Posts
September 13 2009 02:13 GMT
#8
On September 13 2009 10:40 nttea wrote:
Show nested quote +
On September 13 2009 10:37 deconduo wrote:
If each bridge can take any volume of traffic, then I supoose + Show Spoiler +
X=(2/3)L
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
(\o/)  If you want it, you find a way. Otherwise you find excuses. No exceptions.
 /_\   aka Shinbi (requesting a name change since 27/05/09 ☺)
Hypnosis
Profile Blog Joined October 2007
United States2061 Posts
September 13 2009 02:17 GMT
#9
+ Show Spoiler +
x = L+ (L-x).
Science without religion is lame, Religion without science is blind
micronesia
Profile Blog Joined July 2006
United States24670 Posts
September 13 2009 02:28 GMT
#10
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 +
X=(2/3)L
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.
ModeratorThere are animal crackers for people and there are people crackers for animals.
evanthebouncy!
Profile Blog Joined June 2006
United States12796 Posts
September 13 2009 03:10 GMT
#11
On September 13 2009 11:17 Hypnosis wrote:
+ Show Spoiler +
x = L+ (L-x).


u realize that simplifies to x = L right?
Life is run, it is dance, it is fast, passionate and BAM!, you dance and sing and booze while you can for now is the time and time is mine. Smile and laugh when still can for now is the time and soon you die!
Saracen
Profile Blog Joined December 2007
United States5139 Posts
September 13 2009 03:22 GMT
#12
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?
JeeJee
Profile Blog Joined July 2003
Canada5652 Posts
September 13 2009 03:27 GMT
#13
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 +
X=(2/3)L
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..
(\o/)  If you want it, you find a way. Otherwise you find excuses. No exceptions.
 /_\   aka Shinbi (requesting a name change since 27/05/09 ☺)
micronesia
Profile Blog Joined July 2006
United States24670 Posts
September 13 2009 03:30 GMT
#14
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.
ModeratorThere are animal crackers for people and there are people crackers for animals.
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