Jack Climbing the Beanstalk

Date: 6/29/96 at 16:5:3
From: Anonymous
Subject: Jack Climbing the Beanstalk

Jack climbed up the beanstalk at a uniform rate. At 2 P.M. he was 
one-sixth the way up and at 4 P.M. he was three fourths the way up. 
What fractional part of the entire beanstalk had he climbed by 3 P.M. 
At what time did he start climbing? When will he get to the top? How 
long was his trip? And how tall was the beanstalk anyway?

Date: 6/30/96 at 15:54:44
From: Doctor Paul
Subject: Re: Jack Climbing the Beanstalk

Here we go:  

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.  

How tall was the beanstalk?  Unless I'm mistaken (and I certainly 
could be!)  I don't think there's enough information given to find out 
how tall the stalk is.  All we know is the that he climbs 7/24 of the 
tree every hour.  That 7/24 could be as little or as great as it wants 
to be. We'd have to be given more information to get the height.

I think this is a wonderful problem.  Even if there is a way to find 
out how tall the stalk is (and I'm overlooking it) I did the entire 
problem without knowing how tall the stalk was.  It shows how much 
information can be derived from a few simple statements.  If you have 
any future questions, feel more than free to send them in.

-Doctor Paul,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/