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\n" ); document.write( "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.
\n" ); document.write( "Solution 1....
\n" ); document.write( "(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.
\n" ); document.write( "(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.
\n" ); document.write( "(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
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\n" ); document.write( "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.
\n" ); document.write( "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.)
\n" ); document.write( "Solution 2....
\n" ); document.write( "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
\n" ); document.write( "60+0.5t = angle the hour hand makes with 12:00 at t minutes after 2:00
\n" ); document.write( "0+6t = angle the minute hand makes with 12:00 at t minutes after 2:00
\n" ); document.write( "We want the difference between those two angles to be 160 degrees:
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\n" ); document.write( "And, as before, we find the angle between the hands is 160 degrees 40 minutes after 2:00, at 2:40. \n" ); document.write( "