Understanding the Clock Angle Problem

The question asks us to find the smaller angle between the hands of a clock when the time is "10 minutes to 2". First, let's determine the exact time this refers to.

• "10 minutes to 2" means that it is 10 minutes before 2:00.
• This corresponds to the time 1:50.

Now, we need to calculate the positions of the hour hand and the minute hand at 1:50 and find the angle between them.

Calculating the Position of the Minute Hand

The minute hand moves 360 degrees in 60 minutes. This means its speed is:

\(\text{Speed of minute hand} = \frac{360^\circ}{60 \text{ minutes}} = 6^\circ \text{ per minute}\)

At 1:50, the minute hand is exactly at the 50-minute mark on the clock face. The 50-minute mark corresponds to the number 10 on the clock.

The angle of the minute hand is measured clockwise from the 12 o'clock position.

\(\text{Angle of minute hand} = 50 \text{ minutes} \times 6^\circ/\text{minute} = 300^\circ\)

So, the minute hand is at 300 degrees from the 12.

Calculating the Position of the Hour Hand

The hour hand moves 360 degrees in 12 hours. This means its speed is:

\(\text{Speed of hour hand} = \frac{360^\circ}{12 \text{ hours}} = 30^\circ \text{ per hour}\)

The hour hand also moves as the minutes pass. In 60 minutes (1 hour), it moves 30 degrees. So, its speed per minute is:

\(\text{Speed of hour hand} = \frac{30^\circ}{60 \text{ minutes}} = 0.5^\circ \text{ per minute}\)

At 1:50, the time is 1 hour and 50 minutes past 12:00. The total number of minutes past 12:00 is \(1 \times 60 + 50 = 110\) minutes.

The angle of the hour hand is measured clockwise from the 12 o'clock position based on the total minutes past 12:00.

\(\text{Angle of hour hand} = 110 \text{ minutes} \times 0.5^\circ/\text{minute} = 55^\circ\)

So, the hour hand is at 55 degrees from the 12.

Finding the Angle Between the Hands

We have the angle of the minute hand (\(\theta_m\)) and the angle of the hour hand (\(\theta_h\)) from the 12 o'clock position:

• \(\theta_m = 300^\circ\)
• \(\theta_h = 55^\circ\)

The difference between these angles gives the angle between the hands:

\(\text{Difference} = |\theta_m - \theta_h| = |300^\circ - 55^\circ| = |245^\circ| = 245^\circ\)

This difference is one of the angles between the hands. A clock has two angles between the hands: a smaller angle and a larger angle. The sum of these two angles is 360 degrees.

• Larger angle = 245°
• Smaller angle = 360° - Larger angle
• Smaller angle = 360° - 245° = 115°

The question asks for the smaller angle.

Conclusion

At 10 minutes to 2 (1:50), the smaller angle between the hands of the clock is 115°.

The difference is \(|300^\circ - 55^\circ| = 245^\circ\).

The smaller angle is \(360^\circ - 245^\circ = 115^\circ\).

Revision Table: Clock Angle Calculation Summary

Additional Information: Clock Angle Formula

There is a standard formula to find the angle between the hands of a clock at H hours and M minutes:

\(\text{Angle} = |30 \times \text{H} - 5.5 \times \text{M}|\)

Using this formula for H=1 and M=50 (since 1:50 is 1 hour and 50 minutes):

\(\text{Angle} = |30 \times 1 - 5.5 \times 50|\)

\(\text{Angle} = |30 - 275|\)

\(\text{Angle} = |-245|\)

\(\text{Angle} = 245^\circ\)

This formula gives the absolute difference, which might be the larger angle depending on the time. To find the smaller angle, if the result is greater than 180°, subtract it from 360°.

Since \(245^\circ > 180^\circ\), the smaller angle is \(360^\circ - 245^\circ = 115^\circ\).

This confirms the result obtained through step-by-step calculation of individual hand movements. Understanding how each hand moves is key to solving clock angle problems.