SOLUTION: At what time between 2 and 3 O’clock the hands of a clock will make an angle of 160°?

Question 1145926: At what time between 2 and 3 O’clock the hands of a clock will make an angle of 160°?

Answer by greenestamps(13198) (Show Source): You can put this solution on YOUR website!

Note that there will be two answers. At some time between 2 and 3 o'clock the minute hand will be 160 degrees ahead of the hour hand; then at some later time the minute hand will be 200 degrees ahead of the hour hand, making the angle between the hands 360-200 = 160 degrees.

Solution 1....

(1) At 2:00, the minute hand is at 12 and the hour hand is at 2; the minute hand is 60 degrees behind the hour hand.

(2) The hour hand moves 0.5 degrees per minute (30 degrees in 60 minutes); the minute hand move 6 degrees per minute (360 degrees in 60 minutes). So the minute hand moves 5.5 degrees per minute faster than the hour hand.

(3) For the minute hand to get from 60 degrees behind the hour hand (at 2:00) to 160 degrees ahead of the hour hand, the number of minutes required is

So the first time between 2 and 3 o'clock when the angle between the hands is 160 degrees is 40 minutes after 2:00, at 2:40.

Note again there will be a second time when the angle between the hands is 160 degrees. That time is not a "nice" time like 40 minutes; I leave it to you to find it if you need to. (The way the problem is worded, I assume the "nice" answer is the one you were supposed to get.)

Solution 2....

We can formalize the previous solution algebraically. At 2:00, the angle the minute hand makes with 12:00 is 0 degrees; the angle the hour hand makes with 12:00 is 60 degrees. Since the minute hand moves 6 degrees per minute and the hour hand moves 0.5 degrees per minute, we have

60+0.5t = angle the hour hand makes with 12:00 at t minutes after 2:00
0+6t = angle the minute hand makes with 12:00 at t minutes after 2:00

We want the difference between those two angles to be 160 degrees:

And, as before, we find the angle between the hands is 160 degrees 40 minutes after 2:00, at 2:40.
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