At what times between 6 o’clock and 7 o’clock will the hands on a clock be at a right angle?
think about how the clock moves. as the minute hand moves, so does the hour hand. so it may not be as easy as 6:15, 6:30, and 6:45
is 8:15 the same time as quarter after 8:00?
To determine the times between 6 o'clock and 7 o'clock when the hands on a clock form a right angle, we need to consider the movement of the hour hand and the minute hand.
1. Let's start with the minute hand. In one hour, the minute hand completes a full rotation around the clock face, which is 360 degrees. So, the minute hand moves at a rate of 360 degrees per hour.
2. Next, consider the hour hand. In one hour, the hour hand moves a total of 30 degrees. Since there are 12 hours on a clock face, the hour hand moves 360 degrees in 12 hours. This means it moves at a rate of 30 degrees per hour.
3. Now, to find the times when the hands form a right angle, we need to determine the angle between the minute and hour hand. The angle between the hour and minute hand can be calculated as 30H - 11M/2, where H represents the number of hours and M represents the number of minutes.
4. We know that a right angle measures 90 degrees. So, if we set the angle between the hands equal to 90 degrees and solve for H and M, we can find the times.
30H - 11M/2 = 90
Simplifying the equation, we get:
30H - 11M = 180
5. Now, we need to check which values of H and M satisfy this equation and fall between 6 o'clock and 7 o'clock.
We can try the values of H from 6 to 7:
For H = 6, we have:
30(6) - 11M = 180
180 - 11M = 180
-11M = 0
M = 0
So, one possible time is 6 o'clock exactly.
6. Next, we can try the values of M from 0 to 59:
For M = 15, we have:
30H - 11(15)/2 = 180
30H - 165/2 = 180
30H = 180 + 165/2
H = 69/2
This value of H does not fall between 6 and 7, so it is not a valid time.
We continue this process for the remaining values of M until we are out of the time range or find all valid times.
7. After going through all the values of M, we find that no other times between 6 o'clock and 7 o'clock satisfy the equation.
Therefore, the only time between 6 o'clock and 7 o'clock when the hands on a clock form a right angle is at 6 o'clock exactly.
To determine the times between 6 o’clock and 7 o’clock when the hands on a clock will be at a right angle, we need to understand the relationship between the hour and minute hands.
The hour hand moves at a slower pace, moving 30 degrees per hour or 0.5 degrees per minute. The minute hand moves faster, completing a full revolution of 360 degrees in 60 minutes or 6 degrees per minute.
To find when the hour and minute hands are at a right angle, we need to calculate their relative positions. At 6 o’clock, the minute hand is at the 12 o’clock position (0 degrees) while the hour hand is at the 6 o’clock position (180 degrees).
As time progresses, the minute hand moves clockwise, and the hour hand moves at a slower pace. The minute hand's position can be represented by the equation: position = 6 * time. The hour hand's position can be represented by the equation: position = 180 + 0.5 * time.
Now, we need to calculate the time when the difference between the positions of the hour and the minute hand is 90 degrees, as this would indicate a right angle.
Setting up the equation: 180 + 0.5 * time - 6 * time = 90.
Simplifying: -5.5 * time = -90.
Dividing both sides by -5.5: time = -90 / -5.5.
Finding the time: time = 16.36.
Since a clock only has discrete times, we need to round 16.36 to the nearest minute. Thus, the hands will be at a right angle approximately 16 minutes and 22 seconds after 6 o’clock.
To determine the other time when the hands are at a right angle, we can simply subtract the found time from 60 minutes. Thus, a second instance occurs approximately 43 minutes and 38 seconds after 6 o’clock.
Therefore, the times between 6 o’clock and 7 o’clock when the hands on a clock will be at a right angle are around 6:16:22 and 6:43:38.