Through how many degrees does the hour hand in a clock move as the time changes from 3 hours and 12 minutes to 6 hours?
The speed of the hour hand is 360 degrees / 12 hours = 30 degrees per hour.
To find the speed per minute, we divide the hourly speed by 60:
Speed per minute = 30 degrees / 60 minutes = 0.5 degrees per minute.
The time changes from 3 hours and 12 minutes to 6 hours.
First, calculate the duration of this time change.
From 3:12 to 4:00 is 48 minutes (60 – 12).
From 4:00 to 6:00 is 2 hours.
Total duration = 2 hours and 48 minutes.
Convert the total duration into minutes:
2 hours = 2 * 60 = 120 minutes.
Total duration in minutes = 120 minutes + 48 minutes = 168 minutes.
Alternatively, calculate the total time in minutes from 12:00 AM/PM.
3 hours 12 minutes = 3 * 60 + 12 = 180 + 12 = 192 minutes past 12.
6 hours 0 minutes = 6 * 60 + 0 = 360 minutes past 12.
Duration = 360 minutes – 192 minutes = 168 minutes.
The hour hand moves 0.5 degrees for every minute.
Total degrees moved by the hour hand = Duration in minutes * Speed per minute
Total degrees = 168 minutes * 0.5 degrees/minute = 168 * (1/2) = 84 degrees.