When do the hands first form a right angle after 2 o'clock?

Question

Solve in your head. At what exact time after 2:00 do the hour and minute hands first form a right angle (exactly 90 degrees apart)? Aim: under 30 seconds.

Accepted Answer

At 2:00 the hour hand is at 2 	imes 30^\circ = 60^\circ, the minute hand at 0^\circ, so the hour hand leads by 60^\circ. The minute hand closes at 5.5^\circ/min. The separation starts at 60^\circ and shrinks toward 0 (overlap), so it will not reach 90^\circ on that side. Instead the minute hand overtakes the hour hand and then opens a lead; the separation hits 90^\circ once the minute hand is 90^\circ ahead. Total angle the minute hand must gain over the hour hand: 60^\circ to catch up plus 90^\circ to pull ahead, i.e. 150^\circ: t = \frac{150}{5.5} = \frac{300}{11} = 27	frac{3}{11} 	ext{ …

When do the hands first form a right angle after 2 o'clock?

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