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Jack and the Beanstalk
At 2 pm Jack is 4/24 (or 1/6) of the way up and at 4 pm,
two hours later, he is 18/24 (or 3/4) of the way up...
So Jack goes from 4/24 of the way up to 18/24 of the way
up in two hours. That's 14/24 every two hours... reduce
that to 7/12 every two hours.
Now let's divide that rate by two so we know how much of
the stalk he climbs every hour. (7/12) / 2 = 7/24
So Jack climbs 7/24 of the stalk each hour. We know where
he was at 2:00 and how fast he climbs, so now we can figure
out where he was at 3:00.. 4/24 (2:00 position) + 7/24 (his
hourly climbing rate) = 11/24 of the way up at 3:00
Now, when did he start climbing? At 2:00 he was 4/24 of
the way up. At 1:00, he would have been -3/24 of the way
up (if the stalk continued underground).
So we know that he started sometime between 1:00 and 2:00.
Note that it takes Jack ONE HOUR to go from -3/24 to 4/
24. The difference between 4 and -3 is seven so let's divide
the hour into seven equal segments of 8.57 minutes
(THIS IS NOT 8 MINUTES, 57 SECONDS!! IT IS 8 MINUTES, 34.2
SECONDS)
After three of these 8.57 minute segments, Jack would be
at ground zero.
That's 25.71 minutes. Let's convert that to minutes and
seconds: 25 minutes, 42.6 seconds.
Here's what all that means: 25 minutes and 42.6 seconds
after 1:00, Jack leaves the ground.
So Jack started climbing at 1:25:42.6 Now, when will he
get to the top? Well, at 4:00, he was 18/24 of the way up.
At 5:00, he would be at 25/24 (I get this using the predetermined
climbing rate of 7/24 per hour).
Using a little common sense, we can see that he reaches
the top a little before 5:00. Let's figure out exactly when.
6/7 of the way between 4:00 and 5:00, Jack hits the top.
6/7 of 60 minutes is 51.43 minutes.
Convert to minutes and seconds: 51 minutes, 25.8 seconds.
So Jack reaches the top at 4:51:25.8 How long was his trip?
It lasted from 1:25:42 to 4:51:25. That's 3 hours, 25 minutes,
43 seconds.
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