The amount of highway mileage (in thousands of miles) U(t) and R(t) in urban are
ID: 3036379 • Letter: T
Question
The amount of highway mileage (in thousands of miles) U(t) and R(t) in urban areas and rural areas, respectively, can be modeled by the functions U(t) = 12.88t + 625.48 R(t) = -13.38t + 3286.68 where t is the number of years since 1980 (Source: National Highway Traffic Safety Administration). a. Compare U(0) with R(0). What do these values mean in this situation? b. Compare the slopes of the graphs of U and R. What do these slopes mean in this situation? c. Predict when the highway mileage in urban areas will equal the highway mileage in rural areas. d. Predict when the highway mileage in urban areas will be greater than the highway mileage in rural areas.Explanation / Answer
7.(a) We have U(t) = 12.88t+625.48 and R(t) = -13.38t +3286.68. Therefore U(0) = 625.48 and R(0) = 3286.68 so rhat R(0)/U(0) = 3286.68/625.48 = 5.25 (approximately).It means that in 1980, the Rural highway mileage was about 5.25 times the urban highway mileage.
(b) The slope of the graph of U(t) is 12.88 i.e. positive and that of R(t) is – 13.38 i.e. negative. These slopes mean that since 1980, the urban highway mileage has been increasing ( the rate of change being 12.88) and the rural highway mileage has been decreasing ( the rate of change being -13.38).
(c) The urban highway will be equal to the rural highway mileage when U(t) = R(t), I.e. when 12.88t+625.48 =-13.38t +3286.68 or, 12.88t +13.38t = 3286.68-625.48 or, 26.26t = 2661.20 or, t = 2661.20/26.26 = 101.34, say 101 ( on rounding off to the nearest whole number). It means that the urban highway mileage will be equal to the rural highway mileage 101 years after 1980, i.e. by the year 2081.
(d) Since urban highway mileage has been increasing and the rural highway mileage has been decreasingsince 1980, and since the urban highway mileage will be equal to the rural highway mileage by 2081, hence the highway mileage in urban areas will be greater than the highway mileage in rural areas from the year 2082 onwards.
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