In the hour that begins at the time shown, for how long are the hands MORE than 90° apart?

At 3:00 the hands are 90° apart and the minute hand gains 5.5° a minute. They overlap at 16 4/11 min, then the minute hand pulls ahead; the angle passes 90° again when 5.5m − 90 = 90, i.e. m = 32 8/11. It stays above 90° (reaching 180° at 49 1/11 min) until 4:00, when it is 120°. So the time is 60 − 32 8/11 = 27 3/11 min. (32 8/11 is the start time, not the length.)