Chapter 7 Rational Functions - College of the Redwoods

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736 Chapter 7 Rational Functions 60 3 − c = 120 3 + c . Multiply both sides by the common denominator, in this case, (3 − c)(3 + c). [ ] [ ] 60 120 (3 − c)(3 + c) = (3 − c)(3 + c) 3 − c 3 + c Expand each side of this equation. 60(3 + c) = 120(3 − c) 180 + 60c = 360 − 120c This equation is linear (no power of c other than 1). Hence, we want to isolate all terms containing c on one side of the equation. We add 120c to both sides of the equation, then subtract 180 from both sides of the equation. From here, it is simple to solve for c. 60c + 120c = 360 − 180 180c = 180 c = 1. Hence, the speed of the current is 1 mile per hour. It is important to check that the solution satisfies the constraints of the problem statement. • If the speed of the boat in still water is 3 miles per hour and the speed of the current is 1 mile per hour, then the speed of the boat upstream (against the current) will be 2 miles per hour. It will take 30 hours to travel 60 miles at this rate. • The speed of the boat as it goes downstream (with the current) will be 4 miles per hour. It will take 15 hours to travel 60 miles at this rate. Note that the time to travel upstream (30 hours) is twice the time to travel downstream (15 hours), so our solution is correct. Let’s look at another example. ◮ Example 6. A speedboat can travel 32 miles per hour in still water. It travels 150 miles upstream against the current then returns to the starting location. The total time of the trip is 10 hours. What is the speed of the current? Let c represent the speed of the current. Going upstream, the boat struggles against the current, so its net speed is 32−c miles per hour. On the return trip, the boat benefits from the current, so its net speed on the return trip is 32 + c miles per hour. The trip each way is 150 miles. We’ve entered this data in Table 3. Version: Fall 2007
Section 7.8 Applications of Rational Functions 737 Solving d = vt for the time t, d (mi) v (mi/h) t (h) Upstream 150 32 − c ? Downstream 150 32 + c ? Table 3. Entering the given data in a distance, speed, and time table. t = d v . In the first row of Table 3, we have d = 150 miles and v = 32 − c miles per hour. Hence, the time it takes the boat to go upstream is given by t = d v = 150 32 − c . Similarly, upon examining the data in the second row of Table 3, the time it takes the boat to return downstream to its starting location is These results are entered in Table 4. t = d v = 150 32 + c . d (mi) v (mi/h) t (h) Upstream 150 32 − c 150/(32 − c) Downstream 150 32 + c 150/(32 + c) Table 4. Calculating the time to go upstream and return. Because the total time to go upstream and return is 10 hours, we can write 150 32 − c + 150 32 + c = 10. Multiply both sides by the common denominator (32 − c)(32 + c). ( 150 (32 − c)(32 + c) 32 − c + 150 ) = 10(32 − c)(32 + c) 32 + c 150(32 + c) + 150(32 − c) = 10(1024 − c 2 ) We can make the numbers a bit smaller by noting that both sides of the last equation are divisible by 10. 15(32 + c) + 15(32 − c) = 1024 − c 2 Expand, simplify, make one side zero, then factor. 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 7.2 Reducing Rational Funct
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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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