If a clock's hands coincide every 63 minutes, it means that the minute hand will have completed a full revolution (360 degrees) while the hour hand has moved by an angle that brings them together again. Let's calculate how many degrees the hour hand moves in 63 minutes.
In 60 minutes, the hour hand moves 30 degrees (360 degrees divided by 12 hours). So in 1 minute, the hour hand moves 0.5 degrees (30 degrees divided by 60 minutes).
Since the hands coincide every 63 minutes, the minute hand completes 63 full revolutions in that time (360 degrees × 63 = 22,680 degrees). The hour hand would need to cover the same angular distance, so we can set up the equation:
0.5 degrees per minute × T minutes = 22,680 degrees,
where T is the number of minutes it takes for the hour hand to move the required distance.
Simplifying the equation:
0.5T = 22,680, T = 22,680 / 0.5, T = 45,360 minutes.
So, it takes the hour hand 45,360 minutes to coincide with the minute hand. Now we can calculate the clock's time loss per day.
There are 24 hours in a day, and each hour consists of 60 minutes. Therefore, there are 24 * 60 = 1,440 minutes in a day.
The time loss per day is given by the difference between the actual number of minutes in a day (1,440 minutes) and the time it takes for the hour hand to coincide with the minute hand (45,360 minutes):
Time loss per day = 45,360 minutes - 1,440 minutes = 43,920 minutes.
Therefore, the clock loses 43,920 minutes per day, or equivalently, it runs 43,920/60 = 732 hours behind every day.