When do the clock hands overlap between 3 and 4?

Question

At what exact time between 3:00 and 4:00 do the hour and minute hands overlap (angle = 0)?

Accepted Answer

Use the angle formula and set it to zero. With H = 3, 30	imes 3 - 5.5M = 0 \quad\Longrightarrow\quad 5.5M = 90 \quad\Longrightarrow\quad M = \frac{90}{5.5} = \frac{180}{11} = 16	frac{4}{11}\ 	ext{minutes}. So the hands overlap at 3\!:\!16	frac{4}{11}, which is 3:16 and 	frac{4}{11}	imes 60 \approx 22 seconds, about 3:16:22. Intuition: at 3:00 the hour hand is 90^\circ ahead of the minute hand. The minute hand closes the gap at 5.5^\circ per minute (its 6^\circ minus the hour hand's 0.5^\circ), so catching up 90^\circ takes 90 / 5.5 = 16	frac{4}{11} minutes. More generally the hands ove…

When do the clock hands overlap between 3 and 4?

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