The minute hand and hour hand of a clock are at 4 and 12. What is the angle made by these hands at the center? a) 30 degrees b) 60 degrees c) 90 degrees d) 120 degrees
The Correct Answer and Explanation is:
The correct answer is d) 120 degrees.
Explanation:
To find the angle between the hour and minute hands of a clock, we need to understand how the hands move and the positions they occupy at any given time.
- Minute hand movement:
The clock has 60 minutes, and the minute hand completes one full revolution (360 degrees) in 60 minutes. Therefore, for each minute that passes, the minute hand moves: [
\frac{360^\circ}{60} = 6^\circ
] So, for every minute, the minute hand moves 6 degrees. - Hour hand movement:
The hour hand moves 360 degrees in 12 hours. Hence, for each hour, the hour hand moves: [
\frac{360^\circ}{12} = 30^\circ
] So, for every hour, the hour hand moves 30 degrees.
Given the clock positions:
- The hour hand is at 12, which corresponds to 0 degrees.
- The minute hand is at 4, which corresponds to 20 minutes. Since each minute represents 6 degrees of movement, the minute hand moves: [
20 \, \text{minutes} \times 6^\circ/\text{minute} = 120^\circ
]
Finding the angle between the hands:
- The hour hand is at the 12-hour mark, and the minute hand is at the 4-hour mark (20 minutes). The total angle between the hour and minute hands can be calculated by subtracting the position of the hour hand from the position of the minute hand. [
\text{Angle between hands} = 120^\circ – 0^\circ = 120^\circ
]
Thus, the angle between the hour and minute hands when the hour hand is at 12 and the minute hand is at 4 is 120 degrees.
Therefore, the correct answer is d) 120 degrees.