Question

At what time between 4 o'clock and 5 o'clock will the hands of a clock be exactly 10 minutes apart?

Hint:

To solve clock angle problems, use the formula \( \text{Angle} = \left| 30H - 5.5M \right| \) and solve for \( M \).

Answer 1

Step 1: Understanding the problem.
We need to find the time between 4:00 and 5:00 when the hands of the clock are 10 minutes apart. The minute hand moves 6 degrees per minute, and the hour hand moves 0.5 degrees per minute.
Step 2: Calculating the angle between the two hands.
At 4:00, the hour hand is at 120 degrees (since each hour represents 30 degrees, and \( 4 \times 30 = 120 \)).
The angle between the hands can be calculated using the formula: \[ \text{Angle} = \left| 30H - 5.5M \right| \] where \( H \) is the hour and \( M \) is the minute.
We want the angle to correspond to 10 minutes. Since each minute represents 6 degrees: \[ \text{Angle} = 60 \, \text{degrees} \]
Step 3: Solving for the time.
Substitute \( H = 4 \) and set the angle to 60 degrees: \[ \left| 30 \times 4 - 5.5M \right| = 60 \] \[ \left| 120 - 5.5M \right| = 60 \] Solving this gives \( M = 28 \).
Step 4: Conclusion.
The time when the hands are exactly 10 minutes apart is 4:28.