Well IMHO, if it takes five minutes for the alternator to start charging and you only drive it for five minutes to work then it seems to me that you are not charging the battery during the drive and YES that will deplete the battery.
Something is wrong with the alternator. It should start producing charging current within seconds after the engine starts.
I got too many cars!! Insurance eats me alive. Dave
78 Corvette Stingray - 3k
82 242 Turbo Volvo - Manual - 270k
86 300e 5 speed manual - 210k
87 420sel - 240k
89 560sl - 78k
91 420sel - 205k
91 560sel - 85k
94 GMC Suburban - 90k
97 Harley Davidson Heritage Softail - 25k
00 GMC Silverado 1 ton 30k