Finding Clock Hand Angle at Quarter Past Three

The question asks for the angle between the hour and minute hands at "quarter past three". This time is 3:15.

Minute Hand Position

• A clock face is a full circle of 360 degrees.
• There are 60 minutes on a clock face.
• The minute hand moves 360 degrees in 60 minutes, so it moves $360^\circ / 60 = 6^\circ$ per minute.
• At 15 minutes past the hour, the minute hand is pointing at the '3'.
• Its position from the '12' mark is $15 \text{ minutes} \times 6^\circ/\text{minute} = 90^\circ$.

Hour Hand Position

• The hour hand moves 360 degrees in 12 hours, which is $360^\circ / 12 = 30^\circ$ per hour.
• This means the hour hand moves $30^\circ / 60 = 0.5^\circ$ per minute.
• At 3:00 exactly, the hour hand points directly at the '3', which is $3 \times 30^\circ = 90^\circ$ from the '12'.
• By 3:15, the hour hand has moved past the '3' for 15 minutes.
• The additional movement is $15 \text{ minutes} \times 0.5^\circ/\text{minute} = 7.5^\circ$.
• Therefore, the total angle of the hour hand from the '12' mark is $90^\circ + 7.5^\circ = 97.5^\circ$.

Calculating the Angle Difference

• The angle between the hands is the absolute difference between their positions.
• Angle = $| \text{Hour hand angle} - \text{Minute hand angle} |$
• Angle = $| 97.5^\circ - 90^\circ | = 7.5^\circ$.

The angle between the hour hand and the minute hand at quarter past three is $7.5^\circ$.