Question

A clock shows the time as 3:15. What is the angle between the hour and minute hands?

• A. 0°
• B. 7.5°
• C. 30°
• D. 37.5°

Hint:

Use the clock angle formula \( |30H - 5.5M| \) for precision. Remember: minute hand moves 6° per minute, hour hand moves 0.5° per minute. Always take the smaller angle between the hands.

Answer 1

The Correct Option is B

To determine the angle between the clock's hour and minute hands at 3:15, we must first establish the principles of clock hand movement.

1. Core Principles:

- A clock face encompasses 360 degrees across 12 hours.
- Consequently, each hour marker corresponds to \( \frac{360}{12} = 30 \) degrees.
- The minute hand completes a full circle (360 degrees) in 60 minutes, moving at \( \frac{360}{60} = 6 \) degrees per minute.
- The hour hand moves 360 degrees in 12 hours (720 minutes), resulting in a speed of \( \frac{30}{60} = 0.5 \) degrees per minute.

2. Input Data:

- Specified Time: 3:15
- Minute Hand Position: Corresponds to 15 minutes past the hour.
- Hour Hand Position: At 3 hours and 15 minutes past 12.

3. Angle Calculations:

Minute Hand Angle:

Positioned at 15 minutes from the 12 o'clock mark:

\[15 \times 6 = 90^\circ\]

Hour Hand Angle:

At precisely 3:00, the hour hand is at \(3 \times 30 = 90^\circ\).
During the subsequent 15 minutes, the hour hand advances:

\[15 \times 0.5 = 7.5^\circ\]

The total angle for the hour hand is:

\[90 + 7.5 = 97.5^\circ\]

4. Final Angle Determination:

The difference between the two hand angles is:

\[|97.5 - 90| = 7.5^\circ\]

Thus, the angle separating the hour and minute hands is \(7.5^\circ\).

Conclusion:

The angular separation between the clock's hour and minute hands at 3:15 is 7.5 degrees.