Duration of the Game

A basketball playoff game started between 3 pm and 4 pm, and ended between 6 pm and 7 pm. The positions of the minute hand and the hour hand were reversed at the end of the game, compared to the beginning. What was the exact time the game started and ended, and how long was the game? (Try to give exact times, not rounded to the nearest anything.)

Answer:

we know the game started a little after 3:30, since the minute hand was between 6 and 7. Let's make up a variable: let "t" be the number of minutes after 3:00 ; we figure 30 < t < 35. One minute of time is 360/60 = 6 degrees of angle for the minute hand, and 1/12 of that, 1/2 degree, for the hour hand. So at the start, while the minute hand is at 6 t degrees, the hour hand starts at 3:00 (90 degrees) and after t min is at 90 + t/2 degrees. Now at the end of the game, a little after 6:15, the minute hand must be at 90 + t/2 degrees, which is (90 n+ t/2) / 6 = 15 + t/12 min past 6, causing the hour hand to move (90 + t/2) / 12 = 15/2 + t/24 degrees past 6, which has to match up with the original 6 t degrees: 180 + 15/2 + t/24 = 6 t ; mult by 24 and solving gives 4320 + 90 + t = 144 t ; 4410 = 143 t ; Starting time = t = 4410 / 143 = 30 120/143 min past 3 , or 3:30120/143 , approx. 3:30.83916; Ending time = 15 + t/12 = 15 + 2205 / 1716 = 16 489/1716 min past6 , approx 6:16:284965. Time of game = 2 1/2 hrs + 16 489/1716 min - 120/143 min = 2 hrs45 765/1716 min. Approx values in decimal seconds are : Start = 3:30:50.35 , End = 6:16:17.10 , Duration = 2:45:26.75.