ความคิดเห็นที่ 10
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Clearly, m and h are functions of time, so I'll denote them by
m(t) and h(t), respectively.
Now, let both hands start at 0 o'clock at time t%0A 360t = 30 t
330t = 0
t = 0
which is certainly a solution, but there must be other solutions as
well!What happened? Well, if we look closely at our equations,
m(t) = 360t
h(t) = 30t
we can notice something interesting: as t goes from 0 to 12 hrs,
h(t) goes from 0 to 360 degrees, but m(t) goes from 0 to 4320 degrees!
(this means that when the second clock completes one turn, the first
clock will have completed 12 turns).
Clearly, the way we defined m(t), it cannot exceed 360 degrees.
After the first turn, it started from 0 once again, not from 360, and
so on after the second...
This means that m(t) differs from h(t) by a multiple of 360.
Therefore,
m(t) = h(t) + 360*N , where N is an integer
360t = 30t + 360*N
330t = 360*N
360
t = ----- * N
330
or
12
t = -- * N
11
Since 0 <= t< 12 , it follows that 0 <= N < 11
and since N is an integer N = 0 , 1 , 2 , ... , 10
Thus, there are 11 solutions, and the times (in hrs) are
0 hrs or 0:00:00 (the same as 12:00:00)
12/11 hrs or 1:05:27
24/11 hrs or 2:10:54
36/11 hrs or 3:16:21
.
.
120/11 hrs or 10:54:33
We can check that the solutions actually make sense, because after
the two hands meet at 0, the "hours" hand is lagging behind the
"minutes" hand, until the "minutes" hand catches up again to the
"hours" hand in the second turn. We can see that the answer is going
to be a few minutes after one o'clock, because the "minutes" hand
points at 0 o'clock when the "hours" hand points at 1 o'clock,
and it will eventually catch up, since it's faster. And so on for the
third turn, fourth turn, ..., tenth turn.
Now, what happens in the eleventh turn? The "minutes" and "hours"
hands both point back at 0 o'clock,so that the solution is the same
as for the first turn when they met at the beginning. In fact, after
M turns, the solution will be equivalent to that of N = M mod 11
number of turns.
( M mod 11 just means " the remainder when you divide M by 11 ")
If you have any questions, feel free to ask us again. :-)
-Doctor Luis, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
จากคุณ :
araban (araban) 
- [
14 ธ.ค. 49 20:16:52
]
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