What time should I go to bed to wake up at 7:30 AM?
Bedtimes that let you wake up between sleep cycles, not in the middle of one.
Why these times
While you sleep you go through cycles of light, deep and REM sleep that last about 90 minutes on average. Waking up at the end of a cycle usually feels easier than being pulled out of deep sleep, which is what leaves you groggy.
The times above count back whole 90-minute cycles from 7:30 AM and add 15 minutes, roughly what it takes most people to fall asleep. Cycles really last between 70 and 120 minutes and change through the night, so treat these times as a good guide, not an exact rule.
How much should you sleep?
The American Academy of Sleep Medicine recommends at least 7 hours a night for adults and 8 to 10 hours for teenagers. That is why 5 or 6 cycles (7.5 or 9 hours) are the recommended options; 4 cycles is a short night that is fine only once in a while.