There is a flower clock in a certain town and the long hand advances one round in one hour with a fixed speed.
The hour hand stands still first 59 seconds in every minute and advances 1 of 720 round with a fixed speed in last one second.
The long hand and the hour hand have overlapped exactly at 0:00 a.m.
Answer to the following questions.
(1) Find the time when the long hand and the hour hand become right-angled during 0:00 a.m. and 1:00 a.m.
(2) Find the time when the long hand and the hour hand become right-angled during 8:00 a.m. and 9:00 a.m. except exactly at 9:00 a.m.
Answer
(1) 0:16:20 a.m. and 0:49:05 a.m.
(2) 8:27:15 a.m. and 8:59:55 a.m.
Solution
(1) 1st time : The long hand advances 6 degrees in 1 minute (360 degrees / 60 minutes) and the hour hand advances 0.5 degrees in 1 minute (360 degrees × 1/720).
In order to find the time making the the difference of 90 degrees between the long hand and the hour hand, 90 degrees / ( 6 degrees - 0.5 degrees) = 16.3----- minutes.
This indicates that the angle between the long hand and the hour hand at 0:16 00 is 88 degrees ((6 degrees - 0.5 degrees) x 16 minutes = 88).
As for this clock, during the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be wider by 2 degrees more (90 - 88 = 2), it takes 20 seconds (2 degrees / 6 degrees = 1/3 minute = 20 seconds).
Therefore, the time to be found is 0:16:20 a.m.
2nd time : When the long hand advances more than the hour hand by 270 degrees (360 - 90 = 270), the angle is to become 90 degrees again.
The time is calculated as 270 degree / (6 degrees - 0.5 degrees) = 49.0--- minutes.
This indicates that the angle between the long hand and the hour hand at 0:49 00 is 90.5 degrees ((360 degrees - (6 degrees - 0.5 degrees) × 49 minutes = 90.5).
As for this clock, during the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be narrower by 0.5 degrees (90.5 - 90 = 0.5), it takes 5 seconds (0.5 degrees / 6 degrees = 1/12 minute = 5 seconds).
Therefore, the time to be found is 0:49:05 a.m.
(2) The way of thinking is the same as (1).
1st time : The angles between the long hand and the hour hand is 30 degrees × 8= 240 degrees at 8:00.
According to the calculation of (240 - 90) / (6 - 0.5) = 27.2 ... , the angle at 8:27:00 is 240 - (6 - 0.5) × 27 = 91.5 degrees.
During the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be narrower by 1.5 degrees (91.5 - 90 = 1.5), it takes 15 seconds (1.5 degrees / 6 degrees = 1/4 minute = 15 seconds).
Therefore, the time to be found is 8:27:15 a.m.
2nd time : The angles between long and hour is (6 - 0.5) × 59 = 84.5 degrees at 8:59.
During the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be wider by 5.5 degrees (90 - 84.5 = 5.5), it takes 55 seconds (5.5 degrees / 6 degrees = 55/60 minute = 55 seconds).
Therefore, the time to be found is 8:59:55 a.m.
Note : During 59 seconds after 8:59 only the long hand advances.
The angle is 90 degrees at 8:59:55 and it will be 90 degrees again exactly at 9:00.
The hour hand stands still first 59 seconds in every minute and advances 1 of 720 round with a fixed speed in last one second.
The long hand and the hour hand have overlapped exactly at 0:00 a.m.
Answer to the following questions.
(1) Find the time when the long hand and the hour hand become right-angled during 0:00 a.m. and 1:00 a.m.
(2) Find the time when the long hand and the hour hand become right-angled during 8:00 a.m. and 9:00 a.m. except exactly at 9:00 a.m.
Answer
(1) 0:16:20 a.m. and 0:49:05 a.m.
(2) 8:27:15 a.m. and 8:59:55 a.m.
Solution
(1) 1st time : The long hand advances 6 degrees in 1 minute (360 degrees / 60 minutes) and the hour hand advances 0.5 degrees in 1 minute (360 degrees × 1/720).
In order to find the time making the the difference of 90 degrees between the long hand and the hour hand, 90 degrees / ( 6 degrees - 0.5 degrees) = 16.3----- minutes.
This indicates that the angle between the long hand and the hour hand at 0:16 00 is 88 degrees ((6 degrees - 0.5 degrees) x 16 minutes = 88).
As for this clock, during the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be wider by 2 degrees more (90 - 88 = 2), it takes 20 seconds (2 degrees / 6 degrees = 1/3 minute = 20 seconds).
Therefore, the time to be found is 0:16:20 a.m.
2nd time : When the long hand advances more than the hour hand by 270 degrees (360 - 90 = 270), the angle is to become 90 degrees again.
The time is calculated as 270 degree / (6 degrees - 0.5 degrees) = 49.0--- minutes.
This indicates that the angle between the long hand and the hour hand at 0:49 00 is 90.5 degrees ((360 degrees - (6 degrees - 0.5 degrees) × 49 minutes = 90.5).
As for this clock, during the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be narrower by 0.5 degrees (90.5 - 90 = 0.5), it takes 5 seconds (0.5 degrees / 6 degrees = 1/12 minute = 5 seconds).
Therefore, the time to be found is 0:49:05 a.m.
(2) The way of thinking is the same as (1).
1st time : The angles between the long hand and the hour hand is 30 degrees × 8= 240 degrees at 8:00.
According to the calculation of (240 - 90) / (6 - 0.5) = 27.2 ... , the angle at 8:27:00 is 240 - (6 - 0.5) × 27 = 91.5 degrees.
During the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be narrower by 1.5 degrees (91.5 - 90 = 1.5), it takes 15 seconds (1.5 degrees / 6 degrees = 1/4 minute = 15 seconds).
Therefore, the time to be found is 8:27:15 a.m.
2nd time : The angles between long and hour is (6 - 0.5) × 59 = 84.5 degrees at 8:59.
During the following 59 seconds, only the long hand advances.
In order for the angle between the long hand and the hour hand to be wider by 5.5 degrees (90 - 84.5 = 5.5), it takes 55 seconds (5.5 degrees / 6 degrees = 55/60 minute = 55 seconds).
Therefore, the time to be found is 8:59:55 a.m.
Note : During 59 seconds after 8:59 only the long hand advances.
The angle is 90 degrees at 8:59:55 and it will be 90 degrees again exactly at 9:00.
