**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)