[Phys-L] Ex: Re: question
- To:phys-l@mail.phys-l.org
- Sun, Sep 17 at 4:15 PMGentlemen; GENTLEMEN: how could we so readily forget
the classical physics tradition?I refer to the classic Physics device: the
Spherical Cow!
-
-
- Take a list of the contiguous state areas with Alaska and Hawaii from
here:
-
https://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_area
Suppose each state is circular. Its area is then pi*r.^2 The random locations
in the state are equally dispersed, so that we can define an inner circular
area with an outer annular area of each state each of which is equally likely
likely to contain a random visit. The ratio of these two areas is 1:1 and so
the ratio of the radius of the inner circle is 1/sqrt(2) of the outer radius.
Hence the distance of inner circle's circumference to the major circumference
defines the locus of equal distance to the surface of the circular cow - hm...I
mean the average distance to the state border produced by a random visit. this
is 1 - 0.7071 of a state radius - or 0.293*r. So far so good. But the number of
random visits is proportional to the states area, so that the contribution to
the national mean distance to a state border is given by the distance to each
state border times its proportion of the total distribution, which represents
State area / National area (50 states)
This leads to a sum of distance times probability which leads to a mean
distance of 104 km.
For those who have matlab access, this is just these command lines:
>> for i = 1:50 e(i) = 0.293*sqrt(StateArea(i)/pi) .* StateArea(i)/9833517;end;
>>
>> mean(e)
ans =
2.0819
>> sum(e)
ans =
104.0970
>> 104 km...or 65 miles for the 50 states.
- From: John Mallinckrodt <ajm@cpp.edu>
- Date: Mon, 15 May 2023 14:55:06 +0000
In the spirit of the question, I would assume that all places would have
equal
probability of being the drop point.
Unlike some O of M questions, this one has a pretty definite answer and it
involves an easily written double integral that would need to be performed
numerically. I won’t attempt it, but will opine that 50 miles *feels* a little
too small and 100 miles *feels* way too large a little too large.
One must keep in mind that most of the area is taken up by the bigger states.
> On May 15, 2023, at 7:39 AM, Chuck Britton via Phys-l
> <phys-l@mail.phys-l.org> wrote:
>
> My dim memories are that the answer depends strongly on how the ‘random’
> place is chosen. A random angle and radius from a given center will give a
> different answer than will a random latitude and longitude.
>
>> On May 15, 2023, at 10:25 AM, Anthony Lapinski via Phys-l
>> <phys-l@mail.phys-l.org> wrote:
>>
>> I heard about this recently:
>>
>> If you were dropped at a random place in the continental USA, how far would
>> it be to the nearest state border?
>>
>> Hmmm. Tough question! A Fermi question. I did some searching for state
>> areas, perimeters, and centers to get some ideas. Wondering if others can
>> share some insights.
>>
>> --