There is a circular clock.
There is a scale showing the “minutes” which divided the circumference into 60 equally in the dial plate.
Furthermore, the 3rd hand (it is not the second hand) that rotates with fixed speed is supposed.
The center of rotation and the direction of this hand are the same as the hand of the clock.
Answer the following questions.
(2) At a certain time in a.m., the 3rd hand pointed a certain scale.
At this time, the hour hand pointed the 12th scale counterclockwise from the 3rd hand.
Moreover, the long hand pointed the 12th scale clockwise from the 3rd needle.
Find this time.
(3) Afterward at a certain time within 12 hours of the time of (2), the 3rd hand pointed a certain scale.
At this time the hour hand pointed the 12th scale clockwise from the 3rd hand.
Moreover, the long hand pointed the 12th scale counterclockwise from the 3rd hand.
Find this time.
(4) Between the time of (2) and (3), the 3rd hand passed the hour hand 10 times.
Find the time for one time rotation of the 3rd hand.
There is a scale showing the “minutes” which divided the circumference into 60 equally in the dial plate.
(1) Choose all the cases where both the hour hand and the long hand have indicated the scale among the following time.
①0:15 a.m. ②2:36 a.m. ③4:44 a.m. ④8:03 a.m. ⑤10:12 a.m.
①0:15 a.m. ②2:36 a.m. ③4:44 a.m. ④8:03 a.m. ⑤10:12 a.m.
Furthermore, the 3rd hand (it is not the second hand) that rotates with fixed speed is supposed.
The center of rotation and the direction of this hand are the same as the hand of the clock.
Answer the following questions.
(2) At a certain time in a.m., the 3rd hand pointed a certain scale.
At this time, the hour hand pointed the 12th scale counterclockwise from the 3rd hand.
Moreover, the long hand pointed the 12th scale clockwise from the 3rd needle.
Find this time.
(3) Afterward at a certain time within 12 hours of the time of (2), the 3rd hand pointed a certain scale.
At this time the hour hand pointed the 12th scale clockwise from the 3rd hand.
Moreover, the long hand pointed the 12th scale counterclockwise from the 3rd hand.
Find this time.
(4) Between the time of (2) and (3), the 3rd hand passed the hour hand 10 times.
Find the time for one time rotation of the 3rd hand.
Answer
At this time, it is checked whether the hour hand points the scale.
One scale is 360 degrees / 60 = 6 degrees.
The hour hand moves 360 degrees / 12 = 30 degrees in 1 hour.
In 1 minute, it moves 30 degrees / 60 minute = 0.5 degree.
Check the degree that the hour hand moves at the time of ①~⑤ respectively.
① 0.5 degree × 15 minute = 7.5 degrees which is not a multiple of 6, the scale is not pointed.
(2) A problem sentence shows that the hour hand pointed the scale.
According to (1), since the time when the hour hand points are 12, 24, 36, 48, 60 minutes, the time to be found is either of X o’clock and 0 minute, 12 minutes, 24 minutes, 36 minutes, and 48 minutes.
The hour hand points the 12th scale counterclockwise and the long hand points the 12th scale clockwise from the 3rd hand.
Thus, It turns out that the hour hand is 24 scale behind the long hand.
Based on above, the position of the long hand and the hour hand are investigated for 0 minute ~ 48 minutes as it is shown in a lower figure.
The time of possible position of hands in a figure is checked.
(3) Same as (2), the position of the long hand and the hour hand should be shown in the figure to confirm.
According to the problem sentences, it turns out that the hour hand is 24 scale ahead the long hand.
The position of the long hand and the hour hand are investigated for 0 minute ~ 48 minutes as it is shown in a lower figure.
The time of possible position of hands in a figure is checked.
① In case the long hand points 0 minute, the time when the hour hand point 24 minutes is not possible.
② In case the long hand points 12 minutes, the time when the hour hand point 36 minutes is possible.
③ In case the long hand points 24 minutes, the time when the hour hand point 48 minute is not possible.
④ In case the long hand points 36 minutes, the time when the hour hand point 0 minute is not possible.
⑤ In case the long hand points 48 minutes, the time when the hour hand point 12 minutes is not possible.
Therefore, the time to be found is at 7:12.
(4) The time from (2) to (3) is 7:12 - 4:48= 144 minutes.
In 144 minutes, the 3rd hand passes the short hand 10 times.
It means that the 3rd hand rotated 10 round from the position at 4:48 and also it moved to the position at 7:12.
The 3rd hand is a position for 36 minutes at 4:48 and is a position for 24 minutes at 7:12.
The angle is 6 degrees × (60 - 36 + 24) = 6 degrees × 48 minute = 288 degrees.
Thus, the angle by which the 3rd hand moved is to be 360 degrees × 10 + 288 degrees = 3888 degrees.
It took 144 minutes to move 3888 degrees.
The time to move 360 degrees is 144 minutes × 360/3888 = 40/3 minutes = 13 minutes and 20 seconds.
(1) ②，⑤
(2) 4:48
(3) 7:12
(4) 13 minutes and 20 seconds (40/3 minutes)
(2) 4:48
(3) 7:12
(4) 13 minutes and 20 seconds (40/3 minutes)
Solution
(1) Since the scale expresses the minute, at the time of ①~⑤, the long hand always points the scale.
At this time, it is checked whether the hour hand points the scale.
One scale is 360 degrees / 60 = 6 degrees.
The hour hand moves 360 degrees / 12 = 30 degrees in 1 hour.
In 1 minute, it moves 30 degrees / 60 minute = 0.5 degree.
Check the degree that the hour hand moves at the time of ①~⑤ respectively.
① 0.5 degree × 15 minute = 7.5 degrees which is not a multiple of 6, the scale is not pointed.
② 0.5 degree × 36 minute =18 degrees which is a multiple of 6, the scale is pointed.
③ 0.5 degree × 44 minute =22 degrees which is not a multiple of 6, the scale is not pointed.
④ 0.5 degree × 3 minute =1.5 degrees which is not a multiple of 6, the scale is not pointed.
⑤ 0.5 degree × 12 minute =6 degrees which is a multiple of 6, the scale is pointed.
Therefore, the time when both the hour hand and the long hand pointing the scale is ② and ⑤.
③ 0.5 degree × 44 minute =22 degrees which is not a multiple of 6, the scale is not pointed.
④ 0.5 degree × 3 minute =1.5 degrees which is not a multiple of 6, the scale is not pointed.
⑤ 0.5 degree × 12 minute =6 degrees which is a multiple of 6, the scale is pointed.
Therefore, the time when both the hour hand and the long hand pointing the scale is ② and ⑤.
(2) A problem sentence shows that the hour hand pointed the scale.
According to (1), since the time when the hour hand points are 12, 24, 36, 48, 60 minutes, the time to be found is either of X o’clock and 0 minute, 12 minutes, 24 minutes, 36 minutes, and 48 minutes.
The hour hand points the 12th scale counterclockwise and the long hand points the 12th scale clockwise from the 3rd hand.
Thus, It turns out that the hour hand is 24 scale behind the long hand.
Based on above, the position of the long hand and the hour hand are investigated for 0 minute ~ 48 minutes as it is shown in a lower figure.
The time of possible position of hands in a figure is checked.
① In case the long hand points 0 minute, the time when the hour hand point 26 minutes is not possible.
② In case the long hand points 12 minutes, the time when the hour hand point 48 minutes is not possible.
③ In case the long hand points 24 minutes, the time when the hour hand point 0 minute is not possible.
④ In case the long hand points 36 minutes, the time when the hour hand point 12 minutes is not possible.
⑤ In case the long hand points 48 minutes, the time when the hour hand point 24 minutes is possible.
Therefore, the time to be found is at 4:48.
② In case the long hand points 12 minutes, the time when the hour hand point 48 minutes is not possible.
③ In case the long hand points 24 minutes, the time when the hour hand point 0 minute is not possible.
④ In case the long hand points 36 minutes, the time when the hour hand point 12 minutes is not possible.
⑤ In case the long hand points 48 minutes, the time when the hour hand point 24 minutes is possible.
Therefore, the time to be found is at 4:48.
(3) Same as (2), the position of the long hand and the hour hand should be shown in the figure to confirm.
According to the problem sentences, it turns out that the hour hand is 24 scale ahead the long hand.
The position of the long hand and the hour hand are investigated for 0 minute ~ 48 minutes as it is shown in a lower figure.
The time of possible position of hands in a figure is checked.
① In case the long hand points 0 minute, the time when the hour hand point 24 minutes is not possible.
② In case the long hand points 12 minutes, the time when the hour hand point 36 minutes is possible.
③ In case the long hand points 24 minutes, the time when the hour hand point 48 minute is not possible.
④ In case the long hand points 36 minutes, the time when the hour hand point 0 minute is not possible.
⑤ In case the long hand points 48 minutes, the time when the hour hand point 12 minutes is not possible.
Therefore, the time to be found is at 7:12.
(4) The time from (2) to (3) is 7:12 - 4:48= 144 minutes.
In 144 minutes, the 3rd hand passes the short hand 10 times.
It means that the 3rd hand rotated 10 round from the position at 4:48 and also it moved to the position at 7:12.
The 3rd hand is a position for 36 minutes at 4:48 and is a position for 24 minutes at 7:12.
The angle is 6 degrees × (60 - 36 + 24) = 6 degrees × 48 minute = 288 degrees.
Thus, the angle by which the 3rd hand moved is to be 360 degrees × 10 + 288 degrees = 3888 degrees.
It took 144 minutes to move 3888 degrees.
The time to move 360 degrees is 144 minutes × 360/3888 = 40/3 minutes = 13 minutes and 20 seconds.
