Finding Time for Clock Hands at Right Angle Between 5:30 and 6:00

This problem asks us to find the specific time between 5:30 and 6:00 when the hour hand and the minute hand of a clock form a right angle (90 degrees). To solve this type of clock problem, we need to understand the relative speeds of the clock hands.

Clock Hand Speeds

• The minute hand completes a full circle (360 degrees) in 60 minutes. So, its speed is $\frac{360}{60} = 6$ degrees per minute.
• The hour hand completes a full circle (360 degrees) in 12 hours (720 minutes). So, its speed is $\frac{360}{720} = 0.5$ degrees per minute.

Relative Speed of Clock Hands

The minute hand moves faster than the hour hand. The difference in their speeds is the relative speed at which the minute hand gains on the hour hand.

Relative Speed = Speed of minute hand - Speed of hour hand = $6 - 0.5 = 5.5$ degrees per minute.

Calculating Angle Between Clock Hands

The angle between the hour hand and the minute hand at M minutes past H o'clock can be calculated using the formula:

Angle = $|30H - \frac{11}{2}M|$ degrees.

Solving the Right Angle Problem Between 5:30 and 6:00

We want to find the time between 5:30 and 6:00 when the angle between the hands is 90 degrees. Here, H = 5. Let the time be M minutes past 5.

We need the angle to be 90 degrees.

Using the formula: $|30 \times 5 - \frac{11}{2}M| = 90$

$|150 - \frac{11}{2}M| = 90$

This gives us two possibilities:

1. $150 - \frac{11}{2}M = 90$
2. $150 - \frac{11}{2}M = -90$

Let's solve the first case:

$150 - \frac{11}{2}M = 90$

$150 - 90 = \frac{11}{2}M$

$60 = \frac{11}{2}M$

$M = \frac{60 \times 2}{11} = \frac{120}{11}$ minutes

$M = 10\frac{10}{11}$ minutes.

This time is approximately 10.9 minutes past 5, which is 5:10 $\frac{10}{11}$. This is between 5:00 and 5:30, so it's not the time we are looking for (which must be between 5:30 and 6:00).

Now, let's solve the second case:

$150 - \frac{11}{2}M = -90$

$150 + 90 = \frac{11}{2}M$

$240 = \frac{11}{2}M$

$M = \frac{240 \times 2}{11} = \frac{480}{11}$ minutes

To convert this improper fraction to a mixed number, divide 480 by 11:

$480 \div 11 = 43$ with a remainder of $480 - (43 \times 11) = 480 - 473 = 7$.

So, $M = 43 \frac{7}{11}$ minutes.

This time is $43 \frac{7}{11}$ minutes past 5. This is approximately 43.64 minutes past 5, which falls between 5:30 (30 minutes past 5) and 6:00 (60 minutes past 5).

Therefore, the hands of the clock will be at a right angle at $43 \frac{7}{11}$ minutes past 5 within the specified interval.

Conclusion

The time between 5:30 and 6:00 when the hands of a clock are at a right angle (90 degrees) is $43 \frac{7}{11}$ minutes past 5.

Clock Problems Revision Table

Additional Information on Clock Angles

Understanding clock angles is a common topic in aptitude tests. Here are some key points:

• The hands of a clock coincide (0 degrees) once every hour, except between 11 and 12, and 12 and 1, where they coincide only at 12:00. So they coincide 11 times in 12 hours and 22 times in 24 hours.
• The hands of a clock are opposite to each other (180 degrees) once every hour, except between 5 and 6, and 6 and 7, where they are opposite only at 6:00. So they are opposite 11 times in 12 hours and 22 times in 24 hours.
• The hands of a clock are at a right angle (90 degrees) twice every hour, except around 3 and 9. They are at 90 degrees 22 times in 12 hours and 44 times in 24 hours.
• The time gaps between two consecutive 90-degree angles are not exactly 30 minutes because of the relative movement of the hour hand.

Solving clock problems often involves setting up equations based on the angle formula or using the relative speed concept to find the time required for the minute hand to cover a certain angle relative to the hour hand.