Clock Hands at Right Angles in 24 Hours

The question asks us to determine the total number of times the hour and minute hands of a clock form a right angle (90 degrees) over a period of 24 hours. Understanding the relative movement of the clock hands is key to solving this problem.

Understanding Clock Hand Movement and Right Angles

A right angle on a clock face occurs when the hour and minute hands are exactly 15 minute divisions apart. This is because a full circle (360 degrees) has 60 minute divisions, so each division represents $\frac{360 \text{ degrees}}{60 \text{ divisions}} = 6$ degrees. Thus, 15 divisions $\times$ 6 degrees/division = 90 degrees.

• The minute hand moves 360 degrees in 60 minutes, so its speed is $\frac{360}{60} = 6$ degrees per minute.
• The hour hand moves 360 degrees in 12 hours (720 minutes), so its speed is $\frac{360}{720} = 0.5$ degrees per minute.
• The relative speed at which the minute hand gains on the hour hand is $6 - 0.5 = 5.5$ degrees per minute.

Occurrences of Right Angles in 12 Hours

In a standard 12-hour cycle, the clock hands are at right angles multiple times. It might seem like they form a 90-degree angle twice every hour, leading to $2 \times 12 = 24$ times. However, there are specific periods where the count is less due to the unique positions of the hands around 3 o'clock and 9 o'clock.

During a 12-hour period, the hands form a 90-degree angle 22 times, not 24. This is because:

• Between 2 o'clock and 4 o'clock (a 2-hour interval), the hands form a right angle only 3 times instead of the expected 4. The 3 o'clock position is a common point where the hands are at a right angle for both the 2-3 hour interval and the 3-4 hour interval.
• Similarly, between 8 o'clock and 10 o'clock (another 2-hour interval), the hands also form a right angle only 3 times instead of the expected 4. The 9 o'clock position serves as a common right angle for both the 8-9 hour interval and the 9-10 hour interval.

Therefore, we subtract these two 'missing' instances from the total expected count:

Number of times at right angles in 12 hours = $(2 \text{ times/hour} \times 12 \text{ hours}) - 2 \text{ (missed instances)}$

Number of times at right angles in 12 hours = $24 - 2 = 22$ times.

Occurrences of Right Angles in 24 Hours

A 24-hour period is simply two consecutive 12-hour cycles. Since the pattern of the clock hands' relative positions and angles is consistent over each 12-hour cycle, we can calculate the total number of times the hands are at right angles in 24 hours by doubling the count for 12 hours.

Number of times at right angles in 24 hours = Number of times in 12 hours $\times 2$

Number of times at right angles in 24 hours = $22 \times 2 = 44$ times.

Final Answer

Therefore, the hands of a clock are at right angles to each other 44 times in a 24-hour period.