Battery od. Alternator ?

Hello, I'm from Mainz am Rhein and I've had the V8 for a few weeks. The previous owner gave me the tip for this forum and I was able to collect plenty of good information.
Yesterday the following happened and I would like to know the cause:
With an old friend I had been chatting with Burger King in the car and the radio was on all the time. After 1 1/2 hours suddenly a car battery was seen in the small screen. I quickly turned off the music and, of course, the ignition. After about 15 min. I realized that he was "tormented" when he jumped in and almost wanted to accept that I needed a starter.
The previous owner gave the battery the date: 04/2002. Besides, she has a fabulous 100 Ah.
Now I would like to know if the syntom described above is normal and the V8 uses a lot of "juice". Actually, 20 KM distance would have to be enough, about 90 min. - very quiet radio, right? I also paid special attention to the fact that no other consumers were involved. Is it perhaps the alternator? After all, the car is almost exactly 12 years old. How much does a renewal of the LM cost? Or what else can be the cause?
Thank you and wish you a good trip
Mfg.
Mirko