In the unique situation where an 'analog' clock is involved, whether running
or stopped, it turns out that the angle in question is precisely 120 degrees.
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220 degrees
Type your answer here... At 3:15, the hours hand is not pointing at "3". It would have moved forward and is somewhere between 3 and 4. In 12 hrs, the hours hand completes 360 degrees i.e. one full circle. Therefore, in 15 mins (1/4 hr), the hours hand would have moved (1/4) * (360/12) = 7.5 degrees. (In this example, the hours hand is 7.5 degrees from "3") The minutes hand at 3:15 is ponting at "3" on the clock. Therefore, the angle between the hands of a clock at 3:15 is 7.5 degrees.
925
This problem can be solved as follows: The angle Ah of the hour hand of a clock, measured from the position at noon or midnight when the hour and minute hands exactly coincide, is Ah = (360 degrees/12 hours)th, where th is the time in hours, including fractions of hours, because the hour hand moves the entire 360 degrees around the clock in 12 hours. Similarly, the angle Am of the minute hand = (360 degrees/60 minutes)tm, where tm is the time in minutes only, including fractions of minutes. The stated time is 3 + 40/60 + 20/3600 hours = 3.672222... hours and the angle is therefore about 110. 11666666... degrees, using the formula above. The time in minutes only is 40 + 20/60 = 40.33333...., so that the angle of the minute hand is 242 degrees. The difference between them is therefore about 131.833..... degrees, or in fraction form 131 and 5/6.
The 3rd angle is 70 degrees which would form a right angle triangle.