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Answer Posted / sutirtha roy chowdhury
The answer is 44 times FOR TRUE OVERLAPS.
As an explanation, I will approach the answer in following
steps.
Notations:
Hour hand:=H minute hand:=M
Assumption: We consider only H and M (and not the hand for
seconds counter) hence overlap of 2 hands.
1) A very tempting answer (which is also given above) would
have been to consider time points such as 12:00 noon, 1:05
am/ pm etc. and so on leading to final value of 24 but
consider the following situation. Suppose that the overlap
happened at 12:00 noon then, since H covers 30 degree angle
in an hour whereas M covers 360 degree, how can the next
overlap happen at 1:05 because then starting from 12:00 H
has moved 30 degree in 1 hour to reach exactly the point
for 1 o clock marker while in the same time M will move
only 360 degree to reach 12 o clock marker agian to make
the time as 1:00 PM and by the time M reaches the point for
1 o clock marker, H will have moved a bit not to make it an
overlap.
Hence it will nullify the answer which is being discussed.
2) Now let us move towards the correct answer. Note that
the angular speeds of the tips of H and M are 30 degree/
Hour and 360 degree / hour respectively.
3) Let us start from a point where both have already
overlapped because if no such point is available then the
question becomes redundant. H will indicate the tip of hour
hand and M will do the same for the minute one.
4) Now had this been a problem of both H and M starting
from the same point, moving on a stratight line with
different speeds, they would have never met again.
5) But here since they are moving on a circle, it is
equivalent to the aforementioned example of straight line
with the end points of the line identified to make them
move round and round in a circle.
6) Let the starting point of overlap be A. After 1 hour M
is 30 degree behind H (Recall the example in <1> ) and if
it is going to overlap with H in next T hours then it has
to cover 30 degree more than H does in this time (T hours).
Since M cover (360-30=)330 degree more in 1 hour, it will
take (30/330=) 1/11 hour for the next overlap.
7) so starting from the point A of first overlap it has
taken (1+ 1/11=) 12/11 hour for another overlap to happen.
So ein 12/11 hour we have had 2 overlaps and hence in 24
hours we will have (2 * (11/12) * 24)= 44 overlaps (unitary
method).
8) Note that the point A of first overlap is an arbitraty
choice.
| Is This Answer Correct ? | 5 Yes | 45 No |
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