SOLUTION: At what time after 12:00 noon will the hour hand and the minute hand of the clock first form an angle of 120 degree?

Question 89779: At what time after 12:00 noon will the hour hand and the minute hand of the clock first form an angle of 120 degree?
Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
Hi,

At midnight the angle between the hands is 0. Half an hour after this, the angle will be 165 (the hour hand will have moved 15 degrees, and the minute hand will have moved 180 degrees, 180-15=165).

Because the hands movement is continuous (there are no jumps) then to get from 1200(0 degrees), to 1230(165 degrees) the angle must have moved through 120, so the answer is somewhere between 0 and 30 minutes.

It takes the hour hand 720 minutes to get round the clock, and it takes the minute hand 60 minutes to get round the clock, so after x minutes, the hour hand has moved 360*x/720=x/2 degrees, and the minute hand has moved 360*x/60=6*x degrees. Because the first solution lies between 0 and 30 minutes, we know that the minute hand must be after the hour hand, so the equation we have to solve is:

minute_hand - hour_hand = 120.

Or rather 6x - 0.5x = 120.

5.5x = 120.

x=240/11

Which roughly is 21 minutes, 49 seconds.

Hope that helps,

Kev

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