The time gap between the two instants, one before and one after 12:00 noon, when the angle between the hour hand and the minute hand is 66°

Question

Question: The time gap between the two instants, one before and one after $12:00$ noon, when the angle between the hour hand and the minute hand is $66^°$ , is
$1.~~~~ 12 ~text{min}.$$2.~~~~ 16~text{min}.$
$3.~~~~ 18 ~text{min}.$ $4.~~~~ 24 ~text{min}.$

My thought: We know that at $12:00$ noon, both the hour and the minute hand are at position $0^°$. Also after $x$ hours of time, the hour hand travels $x / 12$ rotations around the clock. So after $x$ minutes, it travels $x / (60 cdot 12) = x / 720$.
After $x$ minutes of time, the minute hand travels $x / 60$ rotations around the clock.
Now how to proceed the further ? Please help.

Ross Millikan · Answer

The hour hand travels $360^circ$ in $12$ hours, or $720$ minutes. That is $frac 12^circ$ per minute. The minute hand travels $360^circ$ in $60$ minutes or $6^circ$ per minute, so the minute hand gains $frac {11}2$ degrees/minute. To gain $66^circ$ therefore takes $12$ minutes. Similarly, going backwards, the minutes hand loses $66^circ$ in $12$ minutes, so the times are $24$ minutes apart.

Answered by Ross Millikan on February 1, 2021

The time gap between the two instants, one before and one after 12:00 noon, when the angle between the hour hand and the minute hand is 66°

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