time question

**curious444** · June 4th 2009, 03:02 PM

If it is currently 5:15. At what time down to the exact fraction of a second will the minute hand and hour hand meet? thanks!

**skeeter** · June 4th 2009, 04:25 PM

Originally Posted by curious444

If it is currently 5:15. At what time down to the exact fraction of a second will the minute hand and hour hand meet? thanks!

every 5 minute increment on the face of an analog clock is $\frac{\pi}{6}$ radians.

let the 3 o'clock position be the zero position for both hands.

at t = 0 (5:15) the minute hand is at $\theta = 0$ and is moving with speed $\omega = 2\pi$ radians per hr.

at t = 0, the hour hand is at $\phi = \frac{\pi}{3} + \frac{\pi}{24} = \frac{3\pi}{8}$ radians, moving at $\frac{\pi}{6}$ radians per hr.

for t in hours, the two hands coincide when their respective positions are the same, $\theta = \phi$ ...

$0 + 2\pi \cdot t = \frac{3\pi}{8} + \frac{\pi}{6} \cdot t$

$2t = \frac{3}{8} + \frac{t}{6}$

$\frac{11t}{6} = \frac{3}{8}$

$t = \frac{18}{88} = \frac{9}{44}$ hrs

$\frac{9}{44}$ hrs = 12 min + 16.363636... sec

the hands coincide at 5:27 + 16 and $\frac{12}{33}$ sec

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