At what time between 7 and 8 on an analog clock do the hour and minute hand overlap?
At 7:00 the hour hand has gone through 210° , and angle between the two hands is 210°
the hour hand moves 30° per hour or 1/2 ° per minute
while the minute hand moves 6 degrees per minute
so the minute is gaining on the hour hand at the rate of 5 1/2 or 11/2 ° a minute
So to make up the 210° would take 210÷(11/2)
= 420/11 minutes or 38 2/11 minutes or 38 : 10.9 minutes
so the time is 7 : 38 : 10.9
The above becomes a general solution;
at any time between t and t+1 o'clock,
divide t by 11/2 to find the minutes past t o'clock when the hands overlap
To determine the time at which the hour and minute hands overlap on an analog clock between 7 and 8, let's break down the problem:
1. Understand the clock face: An analog clock has 12 hours marked on its face, with each hour representing 30 degrees (360 degrees divided by 12). The minute hand completes a full revolution in 60 minutes, covering 360 degrees. The hour hand takes 12 hours to do a full revolution, covering 360 degrees.
2. Determine the position of the hour hand: Since we want to find the time between 7 and 8, the hour hand will be positioned closer to 8. To calculate the exact position of the hour hand between 7 and 8, we need to know the relationship between the hour hand movement and the minutes:
- In 60 minutes, the hour hand moves 30 degrees (360 degrees divided by 12).
- In 1 minute, the hour hand moves 0.5 degrees (30 degrees divided by 60).
The number of degrees the hour hand moves depends on the number of minutes elapsed since the last whole hour. For example:
- At 7:00, the hour hand will be on the 7 and pointing directly at it (210 degrees).
- At 7:15, 15 minutes have passed, so the hour hand will have moved 7.5 degrees (15 minutes multiplied by 0.5 degrees).
- At 7:30, 30 minutes have passed, so the hour hand will have moved 15 degrees (30 minutes multiplied by 0.5 degrees).
3. Determine the position of the minute hand: The minute hand moves continuously, covering 360 degrees in 60 minutes. We can find its position by calculating the number of degrees it has moved since the beginning of the given hour. For example:
- At 7:00, the minute hand will be directly on the 12, representing 0 degrees.
- At 7:15, the minute hand will be at the 3, representing 90 degrees.
- At 7:30, the minute hand will be at the 6, representing 180 degrees.
- At 7:45, the minute hand will be at the 9, representing 270 degrees.
4. Calculate the overlap time: The hour and minute hands overlap when their position on the clock face is the same. We need to find a time when the positions of the hour and minute hands are equal.
By using the information from steps 2 and 3, we can calculate that the hour and minute hands overlap at approximately 7:38.182 seconds. To calculate the exact time, you can use a mathematical approach or a programming algorithm that takes into account the angles, such as the Atan2 function.
Note: It is important to clarify that the minute hand gradually moves throughout the minute, while the hour hand appears to jump from one hour to the next once it reaches the full hour mark.