A GEOMETRIC SERIES SOLUTION TO THE WORK PROBLEM. or, How to convince your parents that chores are an infinite process

Transcription

1 006 WVU Math Symposium Thursday, October 6 A GEOMETRIC SERIES SOLUTION TO THE WORK PROBLEM or, How to convince your parents that chores are an infinite process Matt Pascal, WVU Mathematics Department Introduction: The Work Problem stated. and have to shovel snow to clear their parents driveway. If usually takes hours to shovel this alone, and can do it in hours alone, how long will it take them to shovel the whole driveway working together? The Recipe: ) Start with a Series for work done (pages & ) o Linear Representation o -D Representation ) Produce a Series for time (page ) o Linear Representation ) Add the Calculus of Geometric Series (pages 5 & 6) o Limit of a Sequence o The Geometric Series o Convergence ) Combine for a delicious solution (page 6)

2 A Linear Representation of the Series for Work Done: Stage One (Suppose this line represents the entire job) In the first hour, does ½ of the job.... and does ¼ of the job. Together they ve done ¾ of the job in hour leaving ¼ of the job undone. Stage Two In the second stage, does ½ of the remaining ¼ (or, 8 of the total) and does ¼ of the remaining ¼ (or, 6 of the total). This is the same ratio of work done, but the time will be shorter. Together they ve done (or 6 ) of the job in stages only 6 is left!

3 A -Dimensional Representation of the Series for Work Done: does ½ of the job in hour does ¼ of the job in hour Together, they ve done ¾ of the job in hour So, ¼ of the job remains Stage One In the second stage, does ½ of the remaining ¼ (or, 8 of the total) and does ¼ of the remaining ¼ (or, 6 of the total). Together, they ve done ( ) of the job in the first two stages. Stage Two Now only 6 of the job remains.

4 A Linear Representation of the Series for Time: So far, we have the following information: The amount of work completed by and is... And it the amount of time it will take them to complete all these stages is... hours. Will they ever be done? Will and ever stop working? Stage one takes one hour (Suppose this line represents time) Stage two takes ¼ of an hour Stage three takes ¼ of ¼ of an hour Stage four takes ¼ of ¼ of ¼ of an hour

5 5 Both of these sums are of the form Calculus to the rescue a ar ar ar ar This is called a Geometric Series and under certain conditions even though it is infinitely long it will be equal to a finite number. To find these conditions, let s stop adding after k terms and assume that it is equal to the finite number, L. So,... a ar ar ar ar... ar k = L. () Basic Algebra tells us that we can multiply both sides of this equation by r, and so this is equivalent to ar ar ar ar If we subtract these two equations, we get a ar ar ar ar k... ar = rl. () ar ar k... ar = L rl () Next, if we combine all of the like terms, they all cancel. Then, we factor out a on the left and factor out L on the right, and equation () becomes k a( r ) = L( r) () Which gives us the final result k a( r L = r ). (5) In our last leap of math, we have to figure out what happens to r k when k becomes larger and larger (mathspeak: k goes to infinity ).

6 6 The only numbers that don t grow to enormous values when we raise them to very large exponents are proper fractions. And, if a fraction between - and is raised to a very large exponent, the resulting fraction is very, very small (mathspeak: it goes to zero ). The conditions are revealed! As long as r is a fraction between - and, we will have a finite value for our infinitely long sum. This phenomenon is called convergence. and will be relieved of their duties as soon as we find the convergence of the two series! Since we used the general case, we can go straight to the solution: The amount of work completed after this process is... = ( 0) = = (so, the job gets done) The number of hours that it takes to do this is ( 0)... = = = =, or hour and 0 minutes.

7 7 Exploration In this workshop, the numbers chosen ( hours and hours) were nice for several different reasons. For one thing, the corresponding rates (½ of the job in one hour and ¼ of the job in one hour) come out to be commonly used fractions. If the fractions are not common, then the associated solution will use the exact same procedure, but the fractional arithmetic will be more sophisticated. In addition, the rates used guaranteed that the two workers didn t finish the job in less than an hour. Finally, the job corresponded to a convergence to the number in the work done series. For exercises in exploring the relationship between the two rates, consider these problems: () What if the work done series doesn t end with convergence to one, and what if the numbers aren t nice like and? Mary and Joe are a team in a pie eating contest. Individually, Mary can eat 6 pies in one hour and Joe can eat pies in one hour. Produce a geometric series solution that will answer this question: How long will it take for them to eat a total of 5 pies? () What if one or both of the workers could be done in an hour? Jack and Joe are bike mechanics and their duty is to build new bikes for the shop floor. Jack can build a bike in 0 minutes and Joe takes 0 minutes. Produce a geometric series solution that will answer this question: How long will it take for them to build one bike if they work together? {To make it interesting, use portions of an hour rather than minutes.} () In the case of and, produce a geometric series for the amount of work that remains after each stage and show that as they work, this series converges to zero. {Here is the start to this series: - ¾ -... } () Produce a geometric series for the solution to the general problem of this type: Worker M requires m hours to complete a task and worker N requires n hours to complete the same task. How long will it take for workers M and N to complete the task if they work together? Look for detailed solutions to these problems very soon at