Basic stocking rate in Kentucky is 3 to 5 goats per acre. I always recommend that producers start with within this range regardless of the number of rotations they are looking utilizing. Part of this is based on issues of fescue summer slump on pastures and the difficulty of keeping them productive during the summer even under good rotations. This is based on research we have conducted in eastern KY on fescue pastures and we have found this to be about correct for an average of 100 lb. goats. I also recommend that people start and go at least one year at the lower stocking rate to better understand how productive the property can be before you increase the rate. With that the 15 acres you mentioned should be able to graze 45 to 75 goats.
I would recommend at least 7 divisions if you can get them and work to move the animals at least weekly. This will seem like an excess in the spring but in the summer when the forage is growing slower it can get tight.