Answer :
Final answer:
To find when the minute and hour hands of a clock are at a right angle between 10 and 11 o'clock, we set up equations based on their rates of movement and solve for the time t after 10 o'clock. This results in two specific times when the hands are exactly 90 degrees apart.
Explanation:
To find at what time between 10 and 11 o'clock the minute and hour hands will be at a right angle, we need to solve for the time when the hands are 90 degrees apart. At 10 o'clock, the hour hand is at 300 degrees (10 hours times 30 degrees per hour). Since the minute hand moves 360 degrees in 60 minutes, or 6 degrees per minute, and the hour hand moves 0.5 degrees per minute (360 degrees in 720 minutes), we can write the following equation where t is the number of minutes after 10 o'clock:
300 degrees + 0.5t = 6t ± 90 degrees
We need to solve for t when the hands are 90 degrees apart - once when the minute hand is ahead of the hour hand, and once when it is behind. Therefore, we solve two equations:
- 300 + 0.5t = 6t - 90 (minute hand ahead)
- 300 + 0.5t = 6t + 90 (minute hand behind)
By solving these equations, we find the two times when the minute and hour hands are at a right angle between 10 and 11 o'clock.
