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What is the first time after five o'clock that the hands of a clock make an angle 84 degrees?
Assume that we're only looking at the hour and minute hand, and not the second hand. It doesn't specify which hands, and not all clocks have a second hand.
There are 360° in a circle, so the minute hand moves 6° per minute, and the hour hand moves 0.1° per minute.
The minute hand starts out at 0°, and the hour hand starts out at 150°, so we have the position functions of the two hands:
- M(t) = 6*t
- H(t) = t/10 + 150
t is the time (in minutes) after 5 o'clock. Now just find the value of t where H(t) - M(t) = 84 {it will first happen as the minute hand is approaching the hour hand}:
t/10 + 150 - 6*t = 84 ---> 66 = 5.9*t ---> t = 66/5.9 = 11.1864407 minutes. So the time is 11 minutes & 11.19 seconds after 5 o'clock.
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