To determine the angle between the hour hand and the minute hand of a clock at a quarter past six, we need to follow a series of logical steps:
1. Calculate the positions of the hour and minute hands:
- Minute Hand Position:
The minute hand at 15 minutes (a quarter past the hour) is pointing at the 3 on the clock. Since the clock is divided into 12 hours, with each hour equaling 30 degrees (360 degrees/12 hours), we can infer:
[tex]\[
\text{Angle of minute hand} = 15 \times 6 = 90 \text{ degrees}
\][/tex]
because each minute represents 6 degrees (360 degrees/60 minutes).
- Hour Hand Position:
At exactly 6:00, the hour hand is at the 6 o’clock position, which is 180 degrees.
Each hour represents 30 degrees (360 degrees/12 hours).
For every minute, the hour hand moves 0.5 degrees (30 degrees/60 minutes).
Therefore, an additional 15 minutes past 6 o'clock moves the hour hand:
[tex]\[
\text{Additional degrees for hour hand} = 15 \times 0.5 = 7.5 \text{ degrees}
\][/tex]
Hence, the total position of the hour hand at 6:15 is:
[tex]\[
180 \text{ degrees} + 7.5 \text{ degrees} = 187.5 \text{ degrees}
\][/tex]
2. Calculate the angle between the two hands:
- Subtract the minute hand's angle from the hour hand's angle to find the absolute difference:
[tex]\[
\text{Absolute difference} = |187.5 - 90| = 97.5 \text{ degrees}
\][/tex]
3. Determine if it's the smaller of the two possible angles:
Since the maximum angle between the hands of a clock is 360 degrees, if the angle calculated above is greater than 180 degrees, we would subtract it from 360 degrees to find the smaller angle. However, in this case:
[tex]\[
97.5 \text{ degrees} \leq 180 \text{ degrees}
\][/tex]
Thus, the angle between the hour hand and the minute hand at quarter past six is [tex]\(97.5^\circ\)[/tex].
The correct option is:
[tex]\[
\boxed{97.5^\circ}
\][/tex]
