Find the angle between hour-hand and minute-hand in a clock at quarter to six
Of course the minute hand will be at the 45 minute mark, if we let 12:00 to correspond with 0°, then
the minute hand is at 270°
each hour the hour hand moves 30°, (360/12 = 30)
so at a quarter to six it has moved 180 - (1/4)(3)° = 172.5
so the acute angle between the two hands is 270-172.5 = 97.5°
To find the angle between the hour hand and minute hand at quarter to six, we need to determine the positions of the hour and minute hands first.
Step 1: Understand the time
The time given is "quarter to six," which means it is 15 minutes before 6 o'clock.
Step 2: Calculate the position of the hour hand
Since it is quarter to six, the hour hand will be slightly past 5 o'clock. To find its exact position, we calculate how far it would have moved in 15 minutes.
There are 12 hours on a clock, and each hour is divided into 60 minutes. Therefore, the hour hand moves 360 degrees / 12 hours = 30 degrees per hour.
In 15 minutes, the hour hand would have moved 30 degrees/60 minutes * 15 minutes = 7.5 degrees.
Step 3: Calculate the position of the minute hand
Since it is quarter to six, the minute hand will be at the 9-minute mark (15 minutes before the next hour).
Each minute mark on the clock represents 360 degrees / 60 minutes = 6 degrees.
Therefore, the minute hand at the 9-minute mark would be at 6 degrees * 9 minutes = 54 degrees.
Step 4: Determine the angle between the hour and minute hands
To find the angle between the hour and minute hands, we subtract the position of the hour hand (7.5 degrees) from the position of the minute hand (54 degrees).
54 degrees - 7.5 degrees = 46.5 degrees
So, the angle between the hour hand and minute hand at quarter to six is 46.5 degrees.
To find the angle between the hour hand and the minute hand in a clock at quarter to six, follow these steps:
1. Determine the position of the hour hand:
- At 12 o'clock, the hour hand is at 0 degrees.
- Each hour mark on the clock corresponds to an angle of 30 degrees.
- The hour hand moves 1/60th of a degree per minute (since there are 60 minutes in an hour and 360 degrees in a full revolution of the hour hand).
2. Determine the position of the minute hand:
- At 12 o'clock, the minute hand is at 0 degrees.
- The minute hand moves 6 degrees per minute (since there are 60 minutes in an hour and 360 degrees in a full revolution of the minute hand).
3. Calculate the total angle between the hour hand and the minute hand:
- Take the absolute difference between the positions of the hour and minute hands.
- Adjust for any overlap (if the hour hand moves ahead or behind the minute hand).
Now, let's apply these steps to the given time, which is quarter to six:
- At quarter to six, the hour hand is close to the 6 mark but not exactly on it. We need to estimate its position.
- The time is a quarter (15 minutes) before 6, so the minute hand is at the number 9.
- The minute hand is at 270 degrees (9 x 30 degrees = 270 degrees).
- The hour hand is slightly past the 5, as it has moved for 15 minutes.
- The hour hand moves 1/60th of a degree per minute, so for 15 minutes, it would have moved 15/60th of a degree, which equals 1/4th of a degree (0.25 degrees).
- Therefore, the hour hand will be at 5.25 (5 plus 0.25) on the clock face.
- To calculate the angle between the hour and minute hands, we take the absolute difference between their positions: |270 - 315.25|.
- The difference is 45.25 degrees.
Therefore, the angle between the hour hand and the minute hand at quarter to six is approximately 45.25 degrees.