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December 28th, 2014, 09:26 AM  #1 
Newbie Joined: Dec 2014 From: Austria Posts: 5 Thanks: 0  About the optimal number of kilometers of highways a country should have About the optimal number of kilometers of highways a country should have Supposing "Country A" has exactly the right number of km of highways for its surface, population and number of cars, what is the optimal number of km of highways for the countries B, C and D? Country A:  13 000 km of highways  357 000 km2  80 mil. inhabitants  54 mil. cars Country B:  X km of highways (unknown)  640 000 km2  66 mil. inhabitants  38 mil. cars Country C:  X km of highways (unknown)  238 000 km2  20 mil. inhabitants  4.5 mil. cars Country D:  X km of highways (unknown)  93 000 km2  9.9 mil. inhabitants  3.6 mil. cars 
December 28th, 2014, 02:28 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,497 Thanks: 580 
Insufficient information.

December 28th, 2014, 02:54 PM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Agreed. It might be reasonable to suppose that the optimal number of km of highway would be an increasing function of population, cars, and area, in which case I would suggest that countries C and D can get away with at most 13 000 km of highways. 
December 28th, 2014, 11:09 PM  #4 
Newbie Joined: Dec 2014 From: Austria Posts: 5 Thanks: 0 
So, what is the minimum number of countries like A (about which I know the total length of their highways, surface, population and number of cars) that I need to calculate the right number of km of highways for B, C and D?

December 29th, 2014, 06:10 AM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
The trouble is that it would be hard to get people to agree about what the right amount of highway is for a given country. A person who drives a lot might want more, and an environmentalist might want less. How can you decide? You could take a number of countries as 'ideal' and try to fit some model to their numbers, but you'll get different answers depending on how you do that. Here's a thought. Let's say you used a formula of the form $$ H=c\cdot k^\alpha i^\beta c^\gamma $$ where $\alpha,\beta,\gamma,c$ are constants, $H$ is the number of miles of highway, $k$ is the area in square km, $i$ is the number of individuals, and $c$ is the number of cars. Suppose you have a country with the optimal $H$ which merges with a neighboring country which is identical. The new country has $$ 2H=c\cdot (2k)^\alpha (2i)^\beta (2c)^\gamma $$ so $$ 2=2^\alpha 2^\beta 2^\gamma $$ and hence $\alpha+\beta+\gamma=1,$ if you assume that the combined country will also have the perfect amount of highway. 
December 29th, 2014, 10:01 AM  #6 
Newbie Joined: Dec 2014 From: Austria Posts: 5 Thanks: 0 
From your formula it would appear that I need 4 non identical or multiple like countries to find the four unknowns: c, alpha, beta and gamma. I do not know how to solve such a system but it must have a numerical solution in Mathcad.

December 29th, 2014, 10:20 AM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Well, with the additional constraint you'd need only three for a fit. I'd recommend more than that so you won't overfit.

December 29th, 2014, 10:51 AM  #8 
Newbie Joined: Dec 2014 From: Austria Posts: 5 Thanks: 0  
December 29th, 2014, 11:10 AM  #9 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  
December 29th, 2014, 12:54 PM  #10 
Newbie Joined: Dec 2014 From: Austria Posts: 5 Thanks: 0 
In your formula, I guess the first c is a c' and is different from the second? I will try the leastsquares method. Last edited by simpex; December 29th, 2014 at 01:01 PM. 

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