The hands of a clock are in a straight line but opposite in direction when they are 180 degrees apart. This happens when the hour and minute hands are exactly opposite each other.

Step 1: Understanding the occurrence

In one complete cycle of 12 hours, the hands of the clock are opposite each other 11 times.

Since a day has 24 hours, we need to consider two 12-hour cycles.

Step 2: Total occurrences in a day

Since it happens 11 times in 12 hours, in 24 hours, it will occur:

11 \times 2 = 22 \text{ times}

Final Answer:

The hands of a clock are in a straight line but opposite in direction 22times in a day.

Another method

🕰️ Clock Hand Alignment 🤔

To determine how many times the hands of a clock are in a straight line but opposite in direction, let’s analyze the movement of the hour and minute hands.

*Hour Hand Movement:*

– The hour hand moves 360° in 12 hours, which means it moves 30° per hour.

– Since there are 60 minutes in an hour, the hour hand moves 0.5° per minute.

*Minute Hand Movement:*

– The minute hand moves 360° in 60 minutes, which means it moves 6° per minute.

*Opposite Direction Alignment:*

For the hands to be in a straight line but opposite in direction, they must form a 180° angle. This occurs when the minute hand and hour hand are exactly opposite each other.

Let’s calculate the time it takes for the hands to form a 180° angle:

– The hour hand moves 0.5° per minute, and the minute hand moves 6° per minute.

– The relative speed between the two hands is 5.5° per minute (6° – 0.5°).

– To form a 180° angle, the hands need to move 180° / 5.5° per minute ≈ 32.73 minutes.

Since the hands form a 180° angle every 32.73 minutes, we can calculate the number of times this occurs in a day:

– 24 hours × 60 minutes/hour = 1440 minutes

– 1440 minutes ÷ 32.73 minutes ≈ 44 times

However, this calculation includes the times when the hands overlap (i.e., are in the same position). Since we’re interested in the times when the hands are opposite each other, we need to subtract the number of overlaps.

The hands overlap 22 times a day (every 65.4545 minutes). Therefore, we subtract these overlaps from the total:

44 times – 22 overlaps = 22 times

🎉 The hands of a clock are in a straight line but opposite in direction 22 times a day. 🕰️