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 $\displaystyle \frac{\pi}{6}$ radians.
let the 3 o'clock position be the zero position for both hands.
at t = 0 (5:15) the minute hand is at $\displaystyle \theta = 0$ and is moving with speed $\displaystyle \omega = 2\pi$ radians per hr.
at t = 0, the hour hand is at $\displaystyle \phi = \frac{\pi}{3} + \frac{\pi}{24} = \frac{3\pi}{8}$ radians, moving at $\displaystyle \frac{\pi}{6}$ radians per hr.
for t in hours, the two hands coincide when their respective positions are the same, $\displaystyle \theta = \phi$ ...
$\displaystyle 0 + 2\pi \cdot t = \frac{3\pi}{8} + \frac{\pi}{6} \cdot t$
$\displaystyle 2t = \frac{3}{8} + \frac{t}{6}$
$\displaystyle \frac{11t}{6} = \frac{3}{8}$
$\displaystyle t = \frac{18}{88} = \frac{9}{44}$ hrs
$\displaystyle \frac{9}{44}$ hrs = 12 min + 16.363636... sec
the hands coincide at 5:27 + 16 and $\displaystyle \frac{12}{33}$ sec