6.1. (12 points) How many minutes after 17:00 will the angle between the hour and minute hands be exactly the same again

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6.1. (12 points) How many minutes after 17:00 will the angle between the hour and minute hands be exactly the same again
Answer: $54 \frac{6}{11} \approx 54.55$.

Solution. At 17:00, the angle between the hour and minute hands is $\frac{5}{12} \cdot 360^{\circ}=150^{\circ}$. The next moment when the angle will be the same will occur within the next hour, after the minute hand has overtaken the hour hand. Let this happen after $x$ minutes. Then

$$
\left(\frac{x}{60}-\frac{5+\frac{x}{60}}{12}\right) \cdot 360^{\circ}=150^{\circ}
$$

from which $x=54 \frac{6}{11}$ min.