Chapter 7 Rational Functions - College of the Redwoods

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740 Chapter 7 Rational Functions Thus, Bill is working at a rate of 1/2 report per hour. Note how we’ve entered this result in the first row of Table 6. Similarly, Maria is working at a rate of 1/4 report per hour, which we’ve also entered in Table 6. We’ve let t represent the time it takes them to write 1 report if they are working together (see Table 5), so the following calculation gives us the combined rate. Rate = Work Time = 1 report . t h That is, together they work at a rate of 1/t reports per hour. This result is also recorded in Table 6. w (reports) r (reports/h) t (h) Bill 1 1/2 2 Maria 1 1/4 4 Together 1 1/t t Table 6. Calculating the Rate entries. In our discussion above, we pointed out the fact that rates add. Thus, the equation we seek lies in the Rate column of Table 6. Bill is working at a rate of 1/2 report per hour and Maria is working at a rate of 1/4 report per hour. Therefore, their combined rate is 1/2 + 1/4 reports per hour. However, the last row of Table 6 indicates that the combined rate is also 1/t reports per hour. Thus, 1 2 + 1 4 = 1 t . Multiply both sides of this equation by the common denominator 4t. [ 1 (4t) 2 + 1 [ ] 1 = (4t) 4] t 2t + t = 4, This equation is linear (no power of t other than 1) and is easily solved. 3t = 4 t = 4/3 Thus, it will take 4/3 of an hour to complete 1 report if Bill and Maria work together. Again, it is very important that we check this result. • We know that Bill does 1/2 reports per hour. In 4/3 of an hour, Bill will complete Work = 1 2 reports h × 4 3 h = 2 3 reports. That is, Bill will complete 2/3 of a report. • We know that Maria does 1/4 reports per hour. In 4/3 of an hour, Maria will complete Version: Fall 2007
Section 7.8 Applications of Rational Functions 741 Work = 1 4 reports h That is, Maria will complete 1/3 of a report. × 4 3 h = 1 3 reports. Clearly, working together, Bill and Maria will complete 2/3 + 1/3 reports, that is, one full report. Let’s look at another example. ◮ Example 8. It takes Liya 7 more hours to paint a kitchen than it takes Hank to complete the same job. Together, they can complete the same job in 12 hours. How long does it take Hank to complete the job if he works alone? Let H represent the time it take Hank to complete the job of painting the kitchen when he works alone. Because it takes Liya 7 more hours than it takes Hank, let H + 7 represent the time it takes Liya to paint the kitchen when she works alone. This leads to the entries in Table 7. w (kitchens) r (kitchens/h) t (h) Hank 1 ? H Liya 1 ? H + 7 Together 1 ? 12 Table 7. Entering the given data for Hank and Liya. We can calculate the rate at which Hank is working alone by solving the equation Work = Rate × Time for the Rate, then substituting Hank’s data from row one of Table 7. Rate = Work Time = 1 kitchen H hour Thus, Hank is working at a rate of 1/H kitchens per hour. Similarly, Liya is working at a rate of 1/(H + 7) kitchens per hour. Because it takes them 12 hours to complete the task when working together, their combined rate is 1/12 kitchens per hour. Each of these rates is entered in Table 8. w (kitchens) r (kitchens/h) t (h) Hank 1 1/H H Liya 1 1/(H + 7) H + 7 Together 1 1/12 12 Table 8. Because the rates add, we can write Calculating the rates. 1 H + 1 H + 7 = 1 12 . Version: Fall 2007
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7 Rational Functions In this chapte
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Section 7.1 Introducing Rational Fu
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Section 7.3 Graphing Rational Funct
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Chapter 7 Rational Functions - College of the Redwoods

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