Minute and Hour Hands Align at 4:21:49 PM

t what time after 4.00 p.m. is the minute's hand clock exactly aligned with the hour hand?

To determine the time after 4:00 p.m. when the minute hand is exactly aligned with the hour hand, we need to set up an equation based on the positions of the hour and minute hands.

1. Position of the hour hand:

   • At 4:00 p.m., the hour hand is at the 4.
   • The hour hand moves 0.5 degrees per minute (since it moves 30 degrees per hour and there are 60 minutes in an hour).

2. Position of the minute hand:

   • The minute hand moves 6 degrees per minute (since it moves 360 degrees in 60 minutes).

Let's denote the number of minutes after 4:00 p.m. as ( m ).

• The hour hand's position in degrees after ( m ) minutes is: [ 120 + 0.5m ] (120 degrees because the hour hand is at 4, and each hour is 30 degrees).

• The minute hand's position in degrees after ( m ) minutes is: [ 6m ]

We need to find ( m ) such that the positions of the hour and minute hands are the same: [ 120 + 0.5m = 6m ]

Solving for ( m ): [ 120 = 6m - 0.5m ] [ 120 = 5.5m ] [ m = \frac{120}{5.5} ] [ m \approx 21.818 ]

So, the minute hand and the hour hand will be exactly aligned approximately 21.818 minutes after 4:00 p.m.

To convert 0.818 minutes to seconds: [ 0.818 \times 60 \approx 49.1 \text{ seconds} ]

Therefore, the minute hand and the hour hand will be exactly aligned at approximately 4