In an analog clock, the angle formed by the hour and the minute hands varies constantly.
Questions:
- Which angle θ is obtained when it is 5:20 ?
- Determine when exactly at around 4, the angle will be exactly 90°?
In an analog clock, the angle formed by the hour and the minute hands varies constantly.
Questions:
1. θ = 40°
Solution:
θ from 12 for each number = 360° / 12 = 30°
Hour θ movement for each minute = 30° / 60 = 0.5°
For 20 minutes,
number = 4
θ = 4 (30°) = 120°…
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…
Each number on clock makes 30°.
The hour hand also moves, so for each minute, it moves 30/60 = 0.5 deg. for 20 minutes it moves 10 deg.
At 5.20 , θ is 40°.
At around 4:5:30, θ is 90°.…
1. θ = 40°
Solution:
θ from 12 for each number = 360° / 12 = 30°
Hour θ movement for each minute = 30° / 60 = 0.5°
For 20 minutes,
number = 4
θ = 4 (30°) = 120°
For 5,
θ = 5 (30°) = 150°
Hour θ movement = 20 (0.5°) = 10°
Total = 150° + 10° = 160°
θ difference = 160° - 120° = 40°
2. 4:05 (minute is 5.45 to be exact)
Solution:
θ from 12 for each number = 360° / 12 = 30°
Hour θ movement for each minute = 30° / 60 = 0.5°
θ for each minute = 360° / 60 = 6°
For 4,
θ = 4 (30°) = 120°
Equation for 90°:
90° = [120° + x (0.5°)] - x (6°)
90° = [120° + 0.5°x] - 6°x
6°x - 0.5°x = 120° - 90°
5.5°x = 30°
x = 5.45
Time would be 4:05 (5.45 to be exact)
For the second question, it is obvious there will be 2 possible solutions: the 90° angle can be made with the minute hand before or after the hour hand! But with the way the question has been asked (i.e. "around 4"), there is one solution better than the other...
1. Theta=40°
2. There are two solutions
2a: Between 4:05 and 4:06
2b: between 4:38 and 4:39
Given h=hours and m=minutes, you can write the follwing equations for the angle alpha the minute hand forms with the "vertical" and the angle beta the hour hand forms with the vertical
alpha=6*m
beta= 30*h + 0.5m
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