At what time immediately prior to six o'clock are the hands of the clock exactly opposite to each other?
Give the exact time in hours, minutes and seconds.

The hour hand moves 360 degrees in 24 hours (1/120 degree/sec) and the minute hand moves 360 degrees in 1 hour (1/10 degree/sec). They are at the exact spot at 12 (noon or midnight).

Obviously the minute hand moves faster than the hour hand. If they are to be exactly opposite each other, the minute hand must travel 180, 540, 900,...,180+360n degrees more than the hour hand if they both start at 12.

Letting t to be the time in seconds where they are opposite each other,

(1/10-1/120)*t = 180 + 360n, where n = 0, 1, 2,...
t = 21600/11 + 43200n/11

The one that comes right before 6 o clock is 4:54:33.

Posted by np_rt on 2004-01-12 10:36:01