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# KCC's Quizzes AQQ230 about a clock and angle puzzle

In an analog clock, the angle formed by the hour and the minute hands varies constantly.

Questions:

1. Which angle θ is obtained when it is 5:20 ?
2. Determine when exactly at around 4, the angle will be exactly 90°?

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[edited by: emassa at 1:54 PM (GMT -4) on 29 Mar 2023]
Parents
• 1. Which angle θ is obtained when it is 5:20 ?

Long hand: 20 past is 120 degrees from 12:00
Short hand: is 150 degrees + 1/3 of the way between 5 and 6 (30/3 degrees) = 160 degrees
Angle between them is 40 degrees

2. Determine when exactly at around 4, the angle will be exactly 90°?

I started iterating it...

4:00pm
Long hand: 0
Short Hand: 120
Difference: 120

4:10pm
Long hand: 60
Short Hand: 120 + 30/6 = 5
Difference: 65

4:05pm
Long hand: 5/60*360 = 30
Short Hand: 120 + 30*5/60 = 2.5
Difference: 92.5 Too high - add time

4:06pm
Long hand: 6/60*360 = 36
Short Hand: 120 + 30*6/60 = 3
Difference: 87 Too low - subtract time

At this point we know it is between 4:05 and 4:06.  I iterated some more but this successive approximation was a little painful, so I thought a bit more and came up with the equation I was essentially using.

l=long hand angle
s=short hand angle
m=minutes past

s-l = 90
l=6*m
s=120 +(30*m/60)
So
120 + (0.5*m) - 6*m = 90
-5.5*m = -30
m = 30/5.5 = 5.45 minutes past 4 or 4:05:27

• 1. Which angle θ is obtained when it is 5:20 ?

Long hand: 20 past is 120 degrees from 12:00
Short hand: is 150 degrees + 1/3 of the way between 5 and 6 (30/3 degrees) = 160 degrees
Angle between them is 40 degrees

2. Determine when exactly at around 4, the angle will be exactly 90°?

I started iterating it...

4:00pm
Long hand: 0
Short Hand: 120
Difference: 120

4:10pm
Long hand: 60
Short Hand: 120 + 30/6 = 5
Difference: 65

4:05pm
Long hand: 5/60*360 = 30
Short Hand: 120 + 30*5/60 = 2.5
Difference: 92.5 Too high - add time

4:06pm
Long hand: 6/60*360 = 36
Short Hand: 120 + 30*6/60 = 3
Difference: 87 Too low - subtract time

At this point we know it is between 4:05 and 4:06.  I iterated some more but this successive approximation was a little painful, so I thought a bit more and came up with the equation I was essentially using.

l=long hand angle
s=short hand angle
m=minutes past

s-l = 90
l=6*m
s=120 +(30*m/60)
So
120 + (0.5*m) - 6*m = 90
-5.5*m = -30
m = 30/5.5 = 5.45 minutes past 4 or 4:05:27

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