On the Third Day: Time-Angle problems.       Given the Time, what is the Angle? Example 1: What angle do the hands make at 12:40? Solution: The Minute Hand is 2/3 around the hour.      So the Hour Hand must be 2/3 from 12 to 1.      Since the whole angle in the latter is 30 degrees, 2/3 of 30 is 20. So the angle between the Hour Hand and 1 is 10 degrees. If the Hour Hand were on 2, the hands would form 180 degrees. The angle from 1 to 2 is 30 degrees. So the angle at 12:40 is 180 - 30 - 10 = 140 degrees.   Example 2: Find the angle of the hands at 12:41. Solution: Compare this to the above:       The Minute Hand has moved 6 degrees.       So the Hour Hand has moved (1/12)(6) = 1/2 degree.       So the angle has decreased by 6 - 1/2 = 5 1/2 degrees. Thus at 12:41 the angle is 140 - 5 1/2 = 134 1/2 degrees.   The following Theorem can be readily verified using the reasoning of the above problem. THEOREM 2: A change in one minute of time causes a change of 5 1/2 degrees in the angle between the hands. Problem: 1. Find the angle between the hands at exactly 5:00.  2. Using the Theorem, extrapolate to find the angle at 5:06. (Answers:  150 degrees;   117 degrees)