Question 1188816: Determine the smaller angle (Ac) formed by the hands of the clock at 3:30 PM. Give the angle coterminal with angle Ac such that Ac is greater than or equal to 360 degrees and smaller than 720 degrees.
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
There are 60 minute graduations around the 360 degree clock face, so the minute
graduations are 360/60 = 6 degrees apart. Therefore the minute hand's angular
speed is 6 degrees per minute.
The minute hand rotates 12 times as fast as the hour hand, so the hour hand's
angular speed is 1/12 as fast, or 6/12 = 1/2 = 0.5 degrees/minute.
So the angle between the hands is increasing at 6.0-0.5 = 5.5 degrees per minute.
Every time the hands are exactly together they are 0 degrees or 360 degrees apart.
Notice that we can always add (or subtract) 360 degrees to (or from) any angle
between the hands or between the vertical and either hand without incurring any
difficulty.
We will also use the formulas:
Degrees rotated = (degrees per minute)(number of minutes)
and
Number of minutes = (degrees rotated)/(degrees per minute)
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Determine the smaller angle (Ac) formed by [BETWEEN] the hands of the clock at
3:30 PM. Give the angle coterminal with angle Ac such that Ac is greater than or
equal to 360 degrees and smaller than 720 degrees.
The last time before 3:30 PM that we were certain of the position of the hands
(without doing any calculating) was at exactly 3:00 PM. At that instant, the
minute hand was pointing precisely straight up and the hour hand was pointing
precisely to the right and the angle between them was precisely 90 degrees or
360-90=270 degrees or 270+360=630 degrees
In 30 minutes' time, the angle between the hands changed at the rate of 5.5
degrees per minute by 5.5(30)=165 degrees. So the angle between the hands at
that time was 270+165=435. [Essentially the hands were 435-360=75 degrees apart
at 3:30.]
So 435 degrees is the coterminal angle you were asked for.
Edwin
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
There are many ways to set up this kind of problem for solving. Here is the way I like best....
I measure the number of degrees past "noon" that each of the hands of the clock has moved; then the angle between the hands is the difference of those two angles.
At 3:30, the hour hand has moved 3.5 hours past noon at 30 degrees per hour (360 degrees in 12 hours), so it is 105 degrees past noon.
At 3:30, the minute hand has clearly moved 180 degrees past noon. However, to demonstrate the general solution method, it has moved for 30 minutes at 6 degrees per minute (360 degrees in 60 minutes), so it is 180 degrees past noon.
So the angle between the two hands is 180-105 = 75 degrees.
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