Exceedingly rare today, markers like this were once common along America’s earliest improved highways. F.A.P. was the acronym for “Federal Aid Primary”, a designation used from 1916 until the early 1960s for federally funded primary road projects. Georgia DOT historian Amber Rhea, who is leading an effort to catalogue the extant markers, notes that this one for the Corinth-to-Grantville Road dates from 1965, near the end of the time these markers were used.
Vanishing Georgia: Photographs by Brian Brown 
Hi, I’m a historian with GDOT and I’ve been documenting these! I’ve created publicly accessible Google map with all the known FAP locations. Please add this one!
https://www.google.com/maps/d/edit?mid=124f2elDOcA4adM1VUO2cDagXLLk&ll=32.2620371231215%2C-83.22297574999999&z=6
I was able to locate the plans for this project, and they are dated 1965. This is toward the end of when FAP markers were used. http://mydocs.dot.ga.gov/info/designplans12/DesignPlansLibrary/H003785_0000001_CST.PDF