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GIS Help: distance calculation based on ZIP Code

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Data sources do come up here a bit so I will comment that you can
get line-oriented lists of post office locations and more from the
census. The TIGER files would let you find streets etc but are probably
too big for hosting yourself. IIRC, I had to cobble together the zipcode
locations but the data is available. Probably you need more resolution and
other things like traffic data to make a routing decision but I would mention
the landmarks DB probably does have hospitals listed. Also, distances are short
enough that "as the crow flies" you could probably linearize and use scaled
x*x+y*y instead of the spherical stuff. 

I may have put it up here along with other illustrations of things
you can do with public data,

http://www.spottext.com/marchywka/distroform.cfm

( I would caution this site is not exactly professional but it
may be informative, I am using it for idea testing etc )