This problem is on the practice problems for my math club, but I'm not sure how to solve it.
What's the exact time that the hour and minute hands overlap each other between 3 and 4 o clock on an analog clock?
Thanks!
[Math] Time between 3 and 4 that the hour and minute hands overlap each other
algebra-precalculusproblem solving
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Best Answer
Let $x$ be the number of minutes after 3:00 that the hour and minute hands overlap.
Since the hour hand moves $360^\circ$ twice over a $24$-hour period, so it moves $\frac{360^\circ}{12\times60}=0.5^\circ$ per minute. The hour hand would be at $90^\circ$ at time 3:00, therefore, it would be at $90^\circ+0.5^\circ\times x$ at the time that the hands overlap.
For the minute hand: it moves $360^\circ$ every hour, so it moves $\frac{360^\circ}{60}=6^\circ$ every minute. And since it starts at an angle of $0^\circ$ at 3:00, the minute hand should be at the angle of $6^\circ\times x$ when they overlap.
Now, set them equal to each other, you get:
\begin{align*} 90+0.5x&=6x\\ 180+x&=12x\\ 180&=11x\\ x&=\frac{180}{11} \end{align*}
$\therefore\text{The hour and minute hands overlap each other }180/11\text{ minutes after 3:00}.$