[Phys-L] Fermi question on state borders
- John Sohl via Phys-l <phys-l@mail.phys-l.org>Mon, Nov 27 at 4:20 PMBeing
in the spirit of a Fermi question, I don't think we are **allowed**
to look stuff up. We are supposed to reason our way though.
By "Continental" USA I assume we mean the contiguous 48. Which is close
enough to 50 and we don't need to worry about Alaska. I would consider the
contiguous 48 to be a rectangle. Without looking anything up, I'm guessing
we are about 4,000 km wide by 2,000 km tall.
4k X 2k = 8 M km^2
8 M km^2 divided by 50 states (48 rounded up) is 1.6 times million divided
by ten.
That's 1.6 times 100,000 km^2 or about 2 times ten to the 5 (2E5)
km-squared per state.
The box states are probably rectangles but close enough to a square. The
length of each side of a square is the square root of the area. Thus, each
average box state is square root of 2E5.
In my head, it is easier to do square roots of even powers so take the
square root of 20E4.
Square root of twenty is about four and a half (4 squared is 16, 5 squared
is 25, so guess at 4.5 for the root of 20.) Likewise, the square root of
1E4 is 1E2 = 100.
I now have the average size of one side of a state is 4.5 * 100 or 450 km.
We have made so many approximations that fussing over integrals and the
like is a waste of time. I'm going to assume that, on average, you will be
halfway from one side to the center.
So, half of 450 km = 225 km.
*So my final answer is 200 km with a precision of one significant figure.*
How close am I?
John
- - - -
John E. Sohl, Ph.D.
************************************************************************************************************************
John asked: "How close am I?"
It happens I took a crack at this question when it was posed in May, but I
unfairlyused a list of state areas from Wiki and a splendid solver which
handles lists of numbers etc. (called there 'vectors') in a single operation,
so my assumptions includedonly an assumed shape. (I used a circle, though a
square and low aspect ratio rectangles share a similar property - that their
centroid distance to the nearest perimeter can be divided in two parts, 70% and
30%, the smaller of which defines a point on the locus of a smaller figure
which covers half the state's area.).
This method shifts the focus from country wide to state wide, but still
supposes a uniform probability of finding oneself anywhere within the state as
opposed to the fact that people are clustered in conurbations. Still the
randomness of the location is what the question specifies, so we can run with
it.
John's areal estimate of the lower forty-eight was splendidly close to an
accepted value of 8.08 million kilometers, so merely 1% low.
Using 50 for the lower 48 provides an areal error source of -4%
Using half a central distance to the boundary instead of 30% overestimates
the distance we need by 67%
Suppose we could aggregate an error estimate by multiplying the partial errors
(!!!)This would be 0.99 X 0.96X 1.67 = 1.59 (!!!)so John's estimator
of 200 km times 1/1.59 gives 126 km as an updated estimate of probable distance
from a state border.
Hmmmm... I seem to remember my initial estimate was about 65 km so I now
suspect significant errors in my estimate!