Solution:
Explanation in Detail:
To determine how many times the hour hand and minute hand of a clock form a right angle between 1:00 pm and 10:00 pm, we analyze their angular positions over this period.
1.
Calculating Angular Movements:
-
Minute Hand: The minute hand moves 360 degrees in 60 minutes, so in 9 hours (from 1:00 pm to 10:00 pm), it covers:
\[
360 \text{ degrees/hour} \times 9 \text{ hours} = 3240 \text{ degrees}
\]
-
Hour Hand: The hour hand moves 30 degrees in 60 minutes (or 0.5 degrees per minute), covering:
\[
30 \text{ degrees/hour} \times 9 \text{ hours} = 270 \text{ degrees}
\]
2.
Relative Angular Distance:
- The difference in their angular positions over 9 hours is:
\[
3240 \text{ degrees (minute hand)} - 270 \text{ degrees (hour hand)} = 2970 \text{ degrees}
\]
3.
Calculating Right Angles:
- A right angle is formed every 180 degrees.
- Therefore, the number of times they form a right angle is:
\[
\frac{2970 \text{ degrees}}{180 \text{ degrees}} = 16.5
\]
Rounding down, they form a right angle 16 times.
4.
Identifying Times of Right Angles:
- The first right angle occurs at 1:21.8181... pm.
- Subsequent right angles occur approximately every 32.7878... minutes until the last right angle at 9:32.7272... pm.
5.
List of Times:
- The hour and minute hands form a right angle at the following times:
- 1:21.8181 pm
- 1:54.5454 pm
- 2:27.2727 pm
- 3:00 pm
- 3:32.7272 pm
- 4:05.4545 pm
- 4:38.1818 pm
- 5:10.9090 pm
- 5:43.6363 pm
- 6:16.3636 pm
- 6:49.0909 pm
- 7:21.8181 pm
- 7:54.5454 pm
- 8:27.2727 pm
- 9:00 pm
- 9:32.7272 pm
Therefore, the hour hand and minute hand of the clock form a right angle 16 times between 1:00 pm and 10:00 pm, as calculated based on their angular movements and the criteria for forming right angles.
