[tempfixit]

Hi and Welcome to AF.

What year vehicle, mileage?

How old is battery?

Sounds to me like the battery is bad, Check the battery posts and cable ends. Boosting it for ten minutes is a good indication that it has a short in the battery cells. Battery voltage should be 12.6V with out vehicle running.

Check to make sure alternator is working properly. Alternator output should be at the 13.6V to 14.5V range.

If you have another battery from a different vehicle remove the old battery and hook up the other vehicle battery.

If you have a multimeter test the vehicle for a battery drain by disconnecting the negative cable and putting one lead from the multimeter on the positive cable that is still hooked up and one lead on the battery cable that is disconnected if the meter says 0 volts no drain, if showing a voltage drain start pulling fuses to figure out where the drain is.

As for the tranny and radio failing at the same time I don't know. Will have to look at diagrams to see if the two are related somehow.