Question

To the nearest second, what is the first time after $12:00$ that the hour hand and the minute hand of a clock are together?

Answer 1

Finding the first time after $12:00$ that the hour hand and the minute hand of a clock are together:

The minute hand rotates $360°$per hour, that is $\frac{360°}{60min}=6°$per minute

The hour hand rotates $\frac{360°}{12}=30°$per hour that is $\frac{30°}{60}=0.5°$per minute

When the degrees moved by the hour and minute hand are equal, they meet.

Let $h$ be the number of hours passed and $m$ be the number of minutes passed such that the hour hand and the minute hand of a clock are together:

$$\begin{gathered} 6°\left(m\right)=0.5°\left(\left(60\times h\right)+m\right) \\ \Rightarrow 12m=60h+m \\ \Rightarrow 11m=60h \\ \Rightarrow m=\frac{60}{11}h \\ =5.4545h \end{gathered}$$

So we have $1h5.4545minutes$.

$$\begin{gathered} 0.4545minutes=0.4545\left(60sec\right) \\ =27.27sec \\ \approx 27sec \end{gathered}$$

Hence, the first time after $12:00$ that the hour hand and the minute hand of a clock are together is $1h5min27s$.