With the death of Arthur Minit, the legendary country of MADADIA's only clock maker, no one else had much experience yet with making clocks. The new church of MADADIA had just been completed and a new clock was needed to fit into the bell tower.
A clock was made by the locals, using the books of Arthur Minit as a guide.The newly made clock was officially put into use when it showed 6 o'clock. But soon it was noticed that the hour-hand and minute-hand had been interchanged and attached to the wrong spindles. The result was that the hour-hand moved with a speed twelve times that of the minute-hand. When a clock maker arrived from another area, a remarkable thing happened: on the moment he inspected the clock, it showed exactly the right time.
If the clock started in the 6 o'clock position, then what was the first moment that it showed the correct time again?
Let x be the number of minutes later the clock is again correct.
The hour hand is going at 12 times its normal rate. Usually it travels at 1/2 degree per minute; now it goes at 6 degrees per minute and so at its first correct positioning it will have traveled 360 + x/2 degrees so
x = (360+x/2) / 6 ; degrees/(degrees/minute)
= 60 + x/12
11x/12 = 60
x = 12*60/11 = 720/11 minutes
This will repeat (be correct) every 720/11 minutes.
That's how often the hour hand is correct.
The minute hand is going at 1/12 its normal rate; the normal is 6 degrees per minute; it's now 1/2 degree per minute.
After x minutes it would have advanced 6x degrees. The next time it's correct it will have moved 6x - 360 degrees.
(6x - 360) / (1/2) = x
12x - 720 = x
11x = 729
x = 720/11 minutes
This, fortuitously, is the same amount of time for the hour hand to be in the right place.
If the clock started showing the correct time at 6:00, it would show the correc time 720/11 minutes later and every 720/11 minutes after that.
720/11 = 65.4545454545455 minutes
when the time would be about 7:05.45....
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Posted by Charlie
on 2024-10-07 20:03:01 |
proposed solution
