Need help diagnosing starting problem

[sshu]

I have a 05 X5 4.4i with 70k miles. The OEM battery is 3 years old. The car never had a starting issue until today in my 3 years of owning it. I made a lot of consecutive short trips today leaving the radio.

-I drove 3 miles, parked and left the radio on for 5 minutes(family left in car)
-drove again 2 miles
-drove another 2 miles and left radio and parking lights on for 5 minutes
-drove again 2 miles and left radio and parking lights on for 5 minutes.

Then car only clicked but would not start. I called AAA and they checked the battery and alternator and said everything was working fine. The car idled for 10 minutes and then drove home. I went back to the car to try to reset the 4x4 light but it only turned once and then started clicking. I have attached the results AAA printed out. They said the battery is still good and the alternator is fine.

Is it the battery, alternator or did I simply drain my battery and it needs to be trickle charged? I often leave my family in the car for 5 minutes with the radio on, am I better off just replacing the battery so we won't get stuck again?

Thanks,

Scott