Some time after 2o’ clock, the angle made between the minute hand and the vertical (12) on a clock is double the angle made between the hour hand and the vertical (12). What time is it?
Hint
What is the rate of travel of each hand on a clock?
Answer
2:24. The hour hand makes an angle of 72 degrees while the minute hand makes an angle of 144 degrees.
At exactly 2:00, the angle made by the hour hand is 60 degrees from vertical. If you let x = the number of minutes past 2:00 that we are looking for, then the hour hand will be at:
60 degrees + x / 60 min* 30 degrees
where x/60 min is the fraction of a full hour that has past, and is therefore the fractional distance that the hour hand has moved closer to the 3. The 30 degrees is the total angle between the 2 and the 3.
The final position of the minute hand will be:
x/30 min * 180 degrees
for the same reasoning as with the hour hand.
Since the minute hand will make twice the angle with vertical, you can set up the equation:
2*(60 + x / 60 * 30) = x / 30 * 180
Solving for x will give you 24 minutes. The time will be 2:24.