Re: [Phys-L] question (distance to border)
- From: John Denker <jsd@av8n.com>
- Date: Wed, 17 May 2023 10:17:17 -0700
I reckon John M. got the right answer.
When I first saw the question, I came to the same answer, for much
the same reasons, although one can estimate the required integrals
without resorting to numerical methods.
Here's the next level of detail:
1) As always, it is important to document assumptions. It seems
reasonable assume that "random drop" means equal probability per
unit area.
2) That means that large states will figure more prominently int
the average. By a lot. Alaska alone is 15% of the US area. Alaska
plus Texas plus California is about 25%.
3) California is long and skinny, so to a good approximation, ignoring
end-effects, nearest means nearest in the skinny direction. So it's a
one-dimensional integral. The width is about 200 miles, so the 1-dimensional
"radius" is 100 miles. The average is ∫ (1-x) dx / ∫ dx which is half the
radius, so 50 miles.
Texas can be approximated as a disk 300 miles in radius. The integral
is ∫ (1-r) dA / ∫ dA ... where the measure of area in polar coordinates
is dA = r dr. That works out to r/3 or 100 miles. That's less than r/2
because most of the area of the disk is closer to the edge than to the
center.
Alaska is bigger, so more like 150 miles to the border.
At this point we know that 100 miles is surely an overestimate for the
nationwide average.
Without thinking about it very hard, one imagines that a lot of area is
tied up in western states where the distance is more like 50 miles. And
then there are a bunch of small states where the distance is nearly zero
on this scale.
Average it all together and 50 miles has got to be pretty close.
This is the best I can do off the top of my head, without writing anything
down or looking anything up.
On 5/15/23 7:55 AM, John Mallinckrodt wrote:
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 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.