Write two times,in digital notation, when clockwise angle between the hands is:
a. Less than one right angle :
b. Less than two right angles :
c. Greater than 3 right angles:
a. 3:05, and 3:10.
b. 5:00, and 6:05.
c. 10:00, and 11:00.
a. Less than one right angle:
Let's assume that the angle at 12 o'clock is 0 degrees.
For the hands to have an angle less than one right angle, the minute hand should be close to the 12 o'clock position and the hour hand should be at a position between 10 and 11.
One possible time that satisfies this condition is:
Hour Hand: At 10
Minute Hand: At 2
Thus, the time is 10:10.
b. Less than two right angles:
To have an angle less than two right angles, the minute hand should be close to the 12 o'clock position and the hour hand should be at a position between 9 and 10.
One possible time that satisfies this condition is:
Hour Hand: At 9
Minute Hand: At 6
Thus, the time is 9:30.
c. Greater than 3 right angles:
To have an angle greater than three right angles, the minute hand should be at 12 o'clock, and the hour hand should be between 2 and 3.
One possible time that satisfies this condition is:
Hour Hand: At 2
Minute Hand: At 12
Thus, the time is 2:00.
To determine the times when the clockwise angle between the hands of a clock meets the given conditions, we need to understand the relationship between the hour hand and the minute hand.
In a standard clock:
- The hour hand completes one full rotation in 12 hours.
- The minute hand completes one full rotation in 60 minutes.
Now let's solve the given conditions:
a. Less than one right angle:
A right angle is equal to 90 degrees. Since the clock has 360 degrees (a full circle), one right angle is a quarter of the clock. The angle between the hands will be less than one right angle if the minute hand is ahead of the hour hand by less than a quarter of the clock.
To calculate this:
1. Divide the clock into 12 equal parts since there are 12 hours.
2. Determine the size of each part by dividing 360 degrees by 12: 360 / 12 = 30 degrees.
3. Calculate how many degrees the hour hand moves for every hour: 30 degrees.
4. Calculate how many degrees the hour hand moves for every minute: 30 degrees / 60 minutes = 0.5 degrees per minute.
5. Determine the difference in degrees between the hour and minute hand: 0.5 degrees/minute * number of minutes.
Taking point 5 into consideration, let's find two times when the angle is less than one right angle.
i. Time 1: Let's assume the clock is at 3:00. In this case, the hour hand points at the 3, which corresponds to 90 degrees. And the minute hand points at the 12, which corresponds to 0 degrees.
- The minute hand is ahead of the hour hand by 0 degrees - 90 degrees = -90 degrees.
- Since the clock is continuously moving clockwise, it's only a matter of time until the minute hand is ahead of the hour hand by less than 90 degrees. The time 3:00 is an example of this.
ii. Time 2: Let's assume the clock is at 3:15. In this case, the hour hand points at the 3, which corresponds to 90 degrees, but now it is 15 minutes (1/4 of an hour) further.
- The minute hand is ahead of the hour hand by (0.5 degrees/minute * 15 minutes) = 7.5 degrees.
- The angle between the hands is 90 degrees - 7.5 degrees = 82.5 degrees, which is less than one right angle.
b. Less than two right angles:
Similar to the previous case, two right angles are equivalent to half of the clock, which is 180 degrees. For the angle between the hands to be less than two right angles, the minute hand should be ahead of the hour hand by less than half of the clock.
To find two times when the angle is less than two right angles, we can follow the same steps as in part a:
i. Time 1: Let's assume the clock is at 9:00. In this case, the hour hand points at the 9, which corresponds to 270 degrees. And the minute hand points at the 12, which corresponds to 0 degrees.
- The minute hand is ahead of the hour hand by 0 degrees - 270 degrees = -270 degrees.
- Since the clock is continuously moving clockwise, it's only a matter of time until the minute hand is ahead of the hour hand by less than 270 degrees. The time 9:00 is an example of this.
ii. Time 2: Let's assume the clock is at 9:30. In this case, the hour hand points at the 9, which corresponds to 270 degrees, but now it is 30 minutes (1/2 of an hour) further.
- The minute hand is ahead of the hour hand by (0.5 degrees/minute * 30 minutes) = 15 degrees.
- The angle between the hands is 270 degrees - 15 degrees = 255 degrees, which is less than two right angles.
c. Greater than three right angles:
To find two times when the angle between the clock hands is greater than three right angles (greater than 270 degrees), we can follow the same steps as before:
i. Time 1: Let's assume the clock is at 12:30. In this case, the hour hand points at the 12, which corresponds to 0 degrees, but now it is ahead by 30 minutes.
- The minute hand is ahead of the hour hand by (0.5 degrees/minute * 30 minutes) = 15 degrees.
- The angle between the hands is 360 degrees (a full circle) + 15 degrees = 375 degrees, which is greater than three right angles.
ii. Time 2: Let's assume the clock is at 6:45. In this case, the hour hand points at the 6, which corresponds to 180 degrees, but now it is ahead by 45 minutes.
- The minute hand is ahead of the hour hand by (0.5 degrees/minute * 45 minutes) = 22.5 degrees.
- The angle between the hands is 180 degrees + 22.5 degrees = 202.5 degrees, which is greater than three right angles.