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Unknown Cache

GPS R You

A cache by UnderTheGun_y2k
Hidden : 11/25/2008
In Iowa, United States
Difficulty:
3 out of 5
Terrain:
1 out of 5

Size: Size: micro (micro)

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Geocache Description:

The cache is NOT located at the above location. You need to solve the puzzle to obtain the cache coordinates.

Most of us use one every time we venture out to locate a cache. Do you even give it a second thought? It just works and provides you with the information you need. Yes, I'm referring to our ubiquitous GPSr. Do you know how one works? After you sign this log, you will have a better appreciation of how the GPS receiver does its job to find its (and your) position. You will probably need a calculator that computes square roots to at least 8 decimal places to solve this puzzle.

Assume you are on a distant planet named Cacheland. Furthermore, assume that there are GPS satellites orbiting this planet which are emitting signals that your GPS receiver can receive to determine your position on the planet's surface. Your GPS receiver measures the distance to the satellites. Your GPS receiver also knows the locations of the satellites very accurately because of the orbital information and time information it receives. This distance measurement places your location on a sphere centered at that satellite with a radius equal to your distance to the satellite. That sphere around the satellite intersects the planet's surface in a circle. The center of the circle is directly beneath the satellite and your position is somewhere on this circle on the planet's surface.

To make the math problem easier, we will assume we are dealing with a flat region of Cacheland and that you are located on the surface of the planet. Also assume that the planet's surface is a plane and the intersection of the satellite sphere and this plane traces out a circle. This eliminates the vertical axis making the problem easier though still challenging to solve. The best part about this is that when you are done, you will know a little bit about what is going on inside that device you hold in your hands while caching.

Your GPS has given you the following information:
Circle Number X Coordinate of circle's
center
Y Coordinate of
circle's center
Radius of circle
1 -106.97515 61.881302 25.0
2 -56.97515 61.881302 40.0
3 -125.1760152 91.881302 60.0

Your task is to take these three circles which have been created by the GPS satellites and find the point where they intersect. The coordinates of this intersection are the coordinates of the real cache here on planet Earth. To get you started on the right foot, the equation for a circle in a plane can be represented by:
(x - xcenter)2 + (y - ycenter)2 = r2;
where r = radius of the circle; and the center of the circle is (xcenter, ycenter).
Once you have the answer for the intersection, convert x and y into DDD MM.MMM to get the actual cache coordinates.
One more thing, most GPS receivers do this calculation once per second in 3 dimensions with the 4 best satellites. Now maybe you will be amazed the next time you pick up that GPSr to find that next new cache.

You can check your answers for this puzzle on Geochecker.com.

Congrats to RonTon for FTF.

Additional Hints (Decrypt)

Hfr pbbeqvangr genafyngvba.
V'z nggenpgrq gb sreebhf zrgnyf.

Decryption Key

A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z

(letter above equals below, and vice versa)

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There are no Trackables in this cache.

 

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Found it 36     Didn't find it 3     Write note 8     Publish Listing 1     Owner Maintenance 1     

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Current Time:
Last Updated: on 3/6/2014 6:15:06 PM (UTC-08:00) Pacific Time (US & Canada) (2:15 AM GMT)
Rendered From:Unknown
Coordinates are in the WGS84 datum