Hints and solutions to
Problem of the day for Friday, February 14
If you make $1,000 every time the hour and minute hands of a clock form a
90-degree angle, how much would you make in 24 hours?
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Let's just make a list, starting from midnight.
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The first time the hands will be at right angles is a little after
12:15 (it's an interesting problem to figure out the exact
time the hands are at right angles, but that is a problem for another
day).
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Next, at about 12:50 (the right angle points in the other direction,
but it still counts!).
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After that, the next two times are about 1:20 and 1:55.
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Then, 2:25 and 3:00. Then 3:30 and 4:05, and so forth.
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Notice that starting from 12:15, the length of time to the next
right angle alternates: 35 minutes, 30 minutes, 35 minutes, 30 minutes.
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So you make $2000 every 65 minutes.
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There are 24 x 60 = 1440 minutes in a day, and divided by 65
minutes
this gives 22.15. In other words, 22 x 65 = 1430, so there are 10 minutes
left over.
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Where do these 10 minutes go? Well, as we said before, it's not
exactly
12:15 when the hands are at right angles, since the hour hand is moving,
too.
So each "65 minute" interval is actually a little longer, and the
discrepancies
add up to 10 minutes by the end of the day.
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So altogether, you'll make $44,000
for the 44 times the hands are at
right angles.