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Chapter 7 Rational Functions 19. Let t represent the time it takes for each trip, and let r represent the speed of the freight train. Then the speed of the passenger train is r + 20. Using the relationship this leads to the following two equations: distance = rate · time, Freight train: 280 = rt Passenger train: 440 = (r + 20)t Solving for t in each equation, it follows that 280 r Now solve this rational equation for r: 280 r = 440 r + 20 = t = 440 r + 20 =⇒ 280(r + 20) = 440r =⇒ 5600 = 160r =⇒ r = 35 mph 21. Let t represent the time it takes for each part of the trip, and let r represent the speed of the boat in still water. Then the actual speed of the boat going downstream is r + 2, and the speed upstream is r − 2. Using the relationship this leads to the following two equations: distance = rate · time, Trip downstream: 5 = (r + 2)t Trip upstream: 2 = (r − 2)t Solving for t in each equation, it follows that Now solve this rational equation for r: 5 r + 2 = 2 r − 2 5 r + 2 = t = 2 r − 2 =⇒ 5(r − 2) = 2(r + 2) =⇒ 3r = 14 =⇒ r = 14 3 mph Version: Fall 2007
Section 7.8 Applications of Rational Functions 23. Let t represent the time it takes for each part of the trip, and let c represent the speed of the current. Then the actual speed of the boat going downstream is 6 + c, and the speed upstream is 6 − c. Using the relationship this leads to the following two equations: distance = rate · time, Trip downstream: 10 = (6 + c)t Trip upstream: 5 = (6 − c)t Solving for t in each equation, it follows that Now solve this rational equation for r: 10 6 + c = 5 6 − c 10 6 + c = t = 5 6 − c =⇒ 10(6 − c) = 5(6 + c) =⇒ 30 = 15c =⇒ c = 2 mph 25. Let t represent the unknown time, r represent Jean’s rate (reports/hour), and s represent Sanjay’s rate. Using the relationship this leads to the following three equations: work = rate · time, Jean: 1 = r(t + 15) Sanjay: 1 = st together: 1 = (r + s) 10 Solve for r and s in the first two equations, and then plug those values into the third equation to obtain ( 1 1 = t + 15 + 1 ) 10 t Now solve this rational equation for t: ( 1 1 = t + 15 + 1 ) 10 =⇒ t(t + 15) = 10t + 10(t + 15) t =⇒ t 2 + 15t = 20t + 150 =⇒ t 2 − 5t − 150 = 0 =⇒ (t − 15)(t + 10) = 0 =⇒ t = 15 hours Version: Fall 2007
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