This would probably work but you'll have contamination with water and other stuff, and grinding the carrot first would help. A better yield would result from using a completely nonpolar solvent that isn't miscible with water, like hexane, toluene, chloroform, etc. Even gasoline. Grind carrot up, do a soak, filter, maybe do a few washes with water in a sep funnel if you have it (a glass gravy separator would also work). Carotenes are located in cell membranes so you may also find some detergent helps if a simple extraction with nonpolar solvent doesn't work.