2. At what moment between 13:00 and 14:00 do the hour and minute hands of a clock form a straight angle $\left(180^{\circ}\right)$?
First solution.

Let this happen $x$ minutes after 13:00.

The large (minute) hand moves $\frac{1}{60}$ of a full circle, or $6^{\circ}$, in one minute.

The small (hour) hand moves $30^{\circ}$ in one hour, so it moves $0.5^{\circ}$ in one minute. (3 points) After $x$ minutes, the large hand has moved $6x$ degrees, and the small hand has moved $\frac{1}{2}x$ degrees.

The hands will form a straight angle $\left(180^{\circ}\right)$ if the angle moved by the large hand is $210^{\circ}$ greater than the angle moved by the small hand (the small hand starts from the position "1", and the large hand starts from the position "12").

(6 points)

![](https://cdn.mathpix.com/cropped/2024_05_30_20b7f5a96a45568d6d08g-3.jpg?height=532&width=481&top_left_y=288&top_left_x=755)

Therefore, we must have

$$
6x = 210 + \frac{1}{2}x
$$

or $x = \frac{420}{11} = 38 \frac{2}{11}$.

The hands will form a straight angle at 13 hours and $38 \frac{2}{11}$ minutes.