Write two times, in digital notation, when the clockwise angle between
the hands is:
a less than one right angle
b less than two right angles
c greater than three right angles
12:01
12:29
12:46
I'm sure you can come up with others. Heck, use a clock if you need to.
a) To find a time when the clockwise angle between the hands is less than one right angle (90 degrees), we need to consider a time where the minute hand is closer to or at the 12 o'clock position, and the hour hand is slightly ahead.
One such time is 12:25.
At this time, the minute hand is at the 5-minute mark, while the hour hand is slightly past the 12 o'clock position, pointing between the 12 and 1. The angle between the hands is less than 90 degrees.
b) To find a time when the clockwise angle between the hands is less than two right angles (180 degrees), we need to consider a time where the minute hand is closer to or at the 6 o'clock position, and the hour hand is ahead.
One such time is 3:40.
At this time, the minute hand is at the 8-minute mark, while the hour hand is at the 3 o'clock position. The angle between the hands is less than 180 degrees.
c) To find a time when the clockwise angle between the hands is greater than three right angles (270 degrees), we need to consider a time where the minute hand is closer to or at the 9 o'clock position, and the hour hand is ahead.
One such time is 6:30.
At this time, the minute hand is at the 6-minute mark, while the hour hand is between the 6 and 7 o'clock positions, pointing closer to the 7. The angle between the hands is greater than 270 degrees.
To find the times when the clockwise angle between the hands of a clock is less than one right angle, less than two right angles, and greater than three right angles, we need to understand the different angles formed by the hour and minute hands.
Let's start with the basics. In a standard 12-hour clock, the hour hand completes a full rotation in 12 hours, while the minute hand completes a full rotation in 60 minutes. Therefore, the ratio between the movements of the hour and minute hands is 1:12, meaning that for every 12 hours the hour hand moves, the minute hand moves a full rotation.
Now, let's address each case step by step:
a) Less than one right angle:
A right angle is equal to 90 degrees. Since the hour hand completes a full rotation in 12 hours (360 degrees) and the minute hand completes a full rotation in 60 minutes (360 degrees), the difference in degrees covered by both hands in one minute can be calculated.
Hour hand: (360 degrees / 12 hours) * (1 hour / 60 minutes) = 0.5 degrees per minute
Minute hand: (360 degrees / 60 minutes) = 6 degrees per minute
To find the angle between the hands, we subtract the degrees covered by the hour hand from the degrees covered by the minute hand:
6 degrees per minute - 0.5 degrees per minute = 5.5 degrees per minute
To find a time when the angle between the hands is less than one right angle, we need an angle less than 90 degrees. Dividing 90 degrees by 5.5 degrees per minute gives:
90 degrees / 5.5 degrees per minute ≈ 16.36 minutes
Therefore, one example of a time when the angle between the hands is less than one right angle (90 degrees) is around 16 minutes and 22 seconds past any hour. For instance, 12:16:22.
b) Less than two right angles:
To find the times when the angle between the hands is less than two right angles (180 degrees), we follow a similar process:
Hour hand: 0.5 degrees per minute
Minute hand: 6 degrees per minute
To find a time when the angle between the hands is less than 180 degrees, we divide 180 degrees by the difference in degrees per minute:
180 degrees / (6 degrees per minute - 0.5 degrees per minute) ≈ 32.73 minutes
Therefore, one example of a time when the angle between the hands is less than two right angles (180 degrees) is around 32 minutes and 44 seconds past any hour. For example, 1:32:44.
c) Greater than three right angles:
To find the times when the angle between the hands is greater than three right angles, we apply the same method:
Hour hand: 0.5 degrees per minute
Minute hand: 6 degrees per minute
To find a time when the angle between the hands is greater than 270 degrees, we divide 270 degrees by the difference in degrees per minute:
270 degrees / (6 degrees per minute - 0.5 degrees per minute) ≈ 49.09 minutes
Therefore, one example of a time when the angle between the hands is greater than three right angles (270 degrees) is around 49 minutes and 5 seconds past any hour. For example, 2:49:05.
Remember, these times are approximate since the angle between the hands is constantly changing throughout the day.