Find the smaller angle between the hands of a clock at 6:45
To find the smaller angle between the hands of a clock at 6:45, we need to find the angles made by the hour hand and the minute hand separately, and then find the difference between those angles.
The minute hand points at 9 on the clock and each minute mark is 6 degrees apart.
At 6:45, the minute hand is 45 minutes or 45 * 6 = 270 degrees apart from the 12 o'clock position (9 - 12 = -3).
The hour hand moves 30 degrees per hour and 0.5 degrees per minute (360 degrees / 12 hours = 30 degrees/hour, 30 degrees / 60 minutes = 0.5 degrees/minute).
At 6:45, the hour hand is 6 hours and 45 minutes apart from the 12 o'clock position.
So, the hour hand is at (6 * 30) + (45 * 0.5) = 180 + 22.5 = 202.5 degrees apart from the 12 o'clock position.
To find the smaller angle, we need to find the difference between these two angles.
| 270 - 202.5 | = | 67.5 | = 67.5 degrees.
Therefore, the smaller angle between the hands of a clock at 6:45 is 67.5 degrees.