Time Angles
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The setup for this puzzler is quite simple. You have an ordinary clock and the time is exactly 3:15 pm.
What is the angle between the two hands of the clock?
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7.5 degrees
Depends on the clock. If the hour hand jerks like a second hand and only moves when the hour changes then the angle is 0.
If the hour hand moves along with the minute hand then the hour hand is 1/4th between the 3 hour mark and the 4 hour mark. Each mark is 30 degrees ( 360 / 12 = 30 ). So the hour hand is 1/4th of 30 degrees away from the 3 hour mark, or 7.5 degrees.
Assuming it works like a normal mechanical clock, 360/12/4 = 7.5 degrees.
7.5 degrees. The 360 degrees of the clock face is divided into 12 sections of 30 degrees; in the quarter hour from 3 to 3.15, the hour hand will move a quarter of the distance between 3 and 4. 30 / 4 = 7.5.
360 degrees in a circle therefore there are 30 degrees for each hour and thus 7.5 degrees for each quarter hour
At 3:15 the angle between the two hands will be 7.5 degrees.
pi/24 Angles are naturally measured in radians.
Mind you, in the real world, the design of the gearing, manufacturing accuracy limitations, and normal wear and tear will ensure that the actual angle between the hands won’t be precisely 7.5 degrees, but something relatively close to that.
Given that a clock you can’t read by eye is pretty useless, and you have to be able to tell what minute it is in an hour, on the minute, the minute hand will be within two degrees of the minute mark (6 degrees between marks, and you’d want to be in the third closest to the mark for readability). Since it’d be rare to manufacture the minute and hour hand gears in seperate processes, you could assume the same tolerances for both; so you have a mean of 7.5 degrees with variations of plus or minus 4 degrees.
7.5 degrees. Each hour is 1/12th of total. At 3:15, the minutes hand will be 1/4 of the way through. So it will be 1/48th of the total. 360 / 48 = 7.5 degrees.
well, the hour hand went 7.5 degrees from its 3h position, but the minutes hand is at the 15 minute, so its 22.5 degrees between the two of them.
97.5
I mean 7.5
If it’s EXACTLY 3:15 pm, the clock is digital so there are no hands and there’s no angle.
If the clock SHOWED 3:15 pm then you had the 7.5 or 0 degrees angle.
As most of you have realized, the angle is 7.5 degrees.
— FOLLOW UP QUESTION —
How many times in a 24 hour period will there be a 7.5 degree angle between the hour and minute hands?
Assuming it’s from 00:00 to 00:00, the hands will overlap 24 times (23 if you want to ignore the overlap at the second 00:00 and go from 00:00:00 to 23:59:59.99999 instead); in each of those overlaps bar the bookends, there will be two occasions when the hands are 7.5 degrees apart, and one each for the bookends, so 22×2 + 2×1 = 46 times.
@Mark: That was my first thought, but you must recall that the overlaps aren’t what we’re checking but the actual time of the actual 7.5 threshold. So, you’ve got to check the actual times of your bookend occurrences, which are, if my math holds, 90/11 (8.1818…) minutes past (and before) the hour.
So 00:08:10 or so is the first occurrence.
And there *is* just one occurrence the first hour. However…
There are not 24 overlaps in 24 hours. There are only 22 due to the movement of the hour hand. so there are 20*2 bookends, plus 2×1 occurrences that “bookend” the change of day, however those are distinctly part of the same 24 hour period, because they happen at approximately 00:08:10 and 23:51:50.
So, distinctly 22 occurrences unless I’m missing something.
If my math holds.
er … 42 if i could add.
@John Armstrong. Pi/24? Surely you mean Tau/12.
http://tauday.com/
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It should be 7.5 degree
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7.5 degrees would happen 48 times on an ordinary clock over a 24 hour period (given that the angle happens on both sides of the hour hand)
In One hour hour hand will move (360/12) ie. 30 degrees so in 15 minutes Hours hand will move 30/4 i.e 7.5degrees
At 3:15 minutes hand will be at 3 ther fore the difference is 7.5