At 3:00am whatis the measure of the central angle of the hands on an analog clock
Doesn't the angle look like this?
http://www.mathsisfun.com/rightangle.html
To determine the measure of the central angle between the hands of an analog clock at 3:00 AM, we need to consider that at this time, both the hour and minute hands are pointing directly at the 3 on the clock face.
Let's break it down step by step:
1. Determine the angle for each hour on the clock face: A clock is a circle, and a circle is divided into 360 degrees. Since there are 12 hours on an analog clock, each hour mark represents an angle of 360 degrees divided by 12, which equals 30 degrees.
2. Calculate the angle for the hour hand: At 3:00 AM, the hour hand is pointing directly at the 3. Since the hour hand moves gradually between the hour marks, we can calculate how far it has moved by multiplying the number of hours past the 3 (which is 3) by the angle for each hour (which is 30 degrees). Therefore, the hour hand has moved 3 hours * 30 degrees per hour = 90 degrees.
3. Calculate the angle for the minute hand: At 3:00 AM, the minute hand is also pointing directly at the 3. Since the minute hand moves gradually around the entire clock face, we can calculate its position by multiplying the fraction of an hour past 3:00 AM by 360 degrees (the total degrees in a circle). At precisely 3:00 AM, no time has passed since the start of the hour, so the minute hand has not moved. Hence, the angle for the minute hand is 0 degrees.
4. Calculate the central angle: To find the central angle between the hour hand and the minute hand, we subtract the angle of the minute hand from the angle of the hour hand. In this case, it is 90 degrees - 0 degrees = 90 degrees.
Therefore, at 3:00 AM, the measure of the central angle between the hands of an analog clock is 90 degrees.