Math A-lympiad: Final 1996-1997

The 'right' way

Introduction

In building junctions, like a footpath, a cycle-path/track, a motorway, a railway or a goods-train-track, one should weight a lot of different factors against eachother. The shortest road is not the cheapest one, most of the time, and it's seldom the most friendly for the environment. Debates on the pro's and con's of the various possibilities normally progress with great difficulty. In this problem we want to try to find alternatives in a quantitative way, with simple mathematical tools.

A foot path

In the city of Waterbroek, a park is situated near the shopping-centre. From the district "Groenzicht", it is just a few-minute walk through the park to the shopping-centre. Appendix 1 contains a map of the park. The park consists of grass, with a couple of flower-beds, a pond and a playing-ground. As you can see, there are a lot of possible walks from A (the district) to B (the shopping-centre) Because everybody has her/his favourite route, grass and plantation are suffering a lot. The municipality has decided to build a footpath in the park. To prevent the pedestrians of chossing alternative routes again, the footpath should also be the (almost) shortest connection. The path cannot go through the flower-beds, the pond and the playing-ground. For you the question: how to let the path run. The footpath will be a gravel-path of 3 metres width. (In the curves, this may vary a little).
    a.Decide on the (or a ) shortest path from A to B and describe the strategy you followed to come to this path.
In fact, the problem for the municipality is more complicated. The underground is not the same everywhere. Depending on the underground, costs can be higher. On sand, the path will cost f50, per square meter, on clay f150,-, and on peat-soil f250,-. (The different undergrounds are indicated on the map in appendix 1). Further, it's possible to move the flower-beds and to build a bridge across the pond. Moving a flower-bed costs f500,- per square meter; a bridge of two meters width (minimum width) costs f2000,- per meter.
    b.Decide on the (or a) cheapest path from A to B.
The pedestrians want to go from A to B as fast as possible; the municipality as cheap as possible.
    c.Decide on an almost ideal path, concerning the wishes. Explain the reasons for chossing this path in a clear way, so that both parties are satisfied.

A motorway

Since years, a discussion is going on about a new to build motorway between Eindhoven and Nijmegen. On the enclosed map, you can see clearly that the present provincial road intersects/crosses through a lot of places. The enormous amount of growing traffic, changed these places into serious bottlenecks. Furthermore, these places have become practically unfit to live in, especially close to the road.

In deciding on the new route for the motorway, there will be a weigh of a lot of poorly comparable criteria. One of these criteria is of course a fast junction. Other demands are about thee distance to civilised area's: not too far away because of industry and economical potency, but not too close either, because of noise pollution, saving nature (nature area's? natural sceneries?), the ground on which the motorway will run (sand, clay), the limitation of the number of necessary "pieces of art" like flyovers (if another junction is intersected) and bridges, expropriation-costs, and short access-routes from the places along the proposed route to the motorway. Also the amount in which this new route relieves the traffic on other routes, could play a role. For you the task to execute a preparatory study, that will play an important role in the decision-making. Write a solid report on the preparatory study, in which at least the following matters arise:

  1. Calculate the traveling-time (by car) from Eindhoven to Nijmegen in the present situation.
  2. A number of study groups is asked to make a proposal for a new route.

    3. One of the study groups is the Nature-Protection-Foundation. The Nature-Protection-Foundation will make a proposal that is most friendly for the environment, but the Nature-Protection-Foundation knows that there are more parties concerned, so they'll keep that in mind in making their variant/proposal.
    The city-counsils of Eindhoven and Nijmegen both have set up two other study groups as well, one study group to come up with an as fast as possible route, and one study group to investigate the possibilities for an as cheap as possible route (the "0-variant").
    Both study groups keep in mind that there are other parties concerned.
  3. Ultimately, there should be a decision on which route it will be. The variant that will be chosen, will probably be a well-balanced variant: this variant will meet the most the criteria that should play a role in choosing a suitable route. This could lead to the situation that one or more new variants can come up to be investigated.

    4. Decide on an arguable/defensible (numerical) weighing of the criteria, and together with that, a suitable advice for the route that should be chosen.
    Try to find out whether this route is very sensitive to your weighing, i.e. whether the route shifts immediately as soon as the mutual weighing of the different criteria is changed a little.
 

GOOD LUCK!

APPENDIX 2

Driving-speeds
 
type of road
`normal' hours
peak hour*
provincial road
in town-area
50 km/h
40 km/h
with service-roads alongside
70 km/h
50 km/h
outside town-area
80 km/h
70 km/h
motorway
maximum speed
(mostly on circular roads around a city)
100 km/h
80 km/h
maximum speed
120 km/h
90 km/h
traffic-light-crossings
chance for red light
30%
70%
waiting-time at red light
1 minute
3 minutes
 

* when the roadsystem is overcrowded
APPENDIX 3

Costs of crossing infrastructure (in millions of guilders)
 
sort of track 
of motorway
altitude with regard to ground level
type of crossing
crossing infrastructure being a 
polder road--provincial road-motorway
lowered
- 6.0 m
flyover
4 mill
8 mill
16 mill
half-lowered
- 3.0 m
flyover
6 mill
12 mill
24 mill
at ground level
+ 1.0 m
flyover
8 mill
16 mill
32 mill
at ground level
+ 1.0 m
tunnel
20 mill
40 mill
80 mill
dike
+ 7.0 m
through-way
7 mill
14 mill
28 mill
 

Costs to build a 2 x 2 lane motorway, hard shoulders included, per kilometer (in millions of guilders)
 
sort of track
costs
at ground level
25 mill/km
dike
40 mill/km
flyover = 20 - 40 meter
80 mill/km
flyover = 100 meter
120 mill/km
half-lowered open tunnel
80 mill/km
completely lowered open tunnel
140 mill/km
tunnel, constructed from ground level
180 mill/km
sailed in and sunk tunnel
190 mill/km
bored tunnel (2 times diameter of 12 meter)
220 mill/km
slip-roads and exits: 4 per junction
15 mill per junction
* extra lane on ground level
5 mill/km