Neela’s clock is broken. The minute hand rotates around the clock correctly, but the hour hand is stuck in the three o’clock position. Suppose Neela first looks at the clock when the hands are aligned and it shows 3:15. She looks at the clock again and sees that the hour and minute hands of the clock form the arms of an angle measuring −105°. Through which of the following could the minute hand have rotated to reach its current position? −465° −375° −435° −95°
The Correct Answer and Explanation is :
To determine through which angle the minute hand has rotated, we need to understand how the clock hands move in relation to each other.
When Neela first looks at the clock at 3:15, the hands are aligned. At 3:15:
- The minute hand is on the 3 (i.e., pointing at the 15-minute mark), and
- The hour hand is stuck at the 3 (the 3 o’clock position), as per the problem.
The clock later shows that the angle between the hour and minute hands is −105°. A negative angle means the minute hand has rotated counterclockwise (backwards). So, we are tasked with figuring out how much the minute hand has moved from the aligned position of 3:15 to this current angle of −105°.
- Relationship Between Hands:
- The minute hand rotates 360° in 60 minutes, meaning it moves 6° per minute (since ( 360^\circ \div 60 = 6^\circ )).
- The hour hand (if functioning) moves 30° per hour, but in this case, it is stuck.
- Finding the Rotation:
- At 3:15, the hands were aligned. After some time, the hands form a −105° angle. For this angle to appear, the minute hand must have moved counterclockwise a total of 105° away from the hour hand (stuck at the 3 o’clock position).
- Calculating the Total Rotation:
- Since the minute hand moves in full revolutions, we need to find the cumulative rotation of the minute hand. We start with −105°, but the minute hand has likely made multiple revolutions to reach that position. We now calculate the possible total rotations:
- −465° (full revolution plus additional movement backwards),
- −375°,
- −435°,
- −95°.
- Conclusion:
The correct answer is −375°. Here’s why: −375° is equivalent to one full counterclockwise rotation (−360°) plus an additional −15°, bringing the minute hand back to a position where it forms a −105° angle with the hour hand, which is stuck at 3.