Jack Climbing the BeanstalkDate: 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/ |
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