377 3.4 Polynomial Functions: Graphs, Applications, and Models x (in mph) 20 30 40 50 60 65 70 y (in feet) 810 1090 1480 1840 2140 2310 2490 Data from Haefner, L., Introduction to Transportation Systems, Holt, Rinehart and Winston. (Modeling) Solve each problem. See Example 8. 99. Highway Design To allow enough distance for cars to pass on two-lane highways, engineers calculate minimum sight distances between curves and hills. The table shows the minimum sight distance y in feet for a car traveling at x miles per hour. Year Amount (billions of $) Year Amount (billions of $) 2008 29.9 2013 32.9 2009 30.6 2014 30.9 2010 34.2 2015 31.4 2011 36.2 2016 31.8 2012 34.5 2017 34.9 Data from U.S. Office of Management and Budget. Which one of the following functions provides the best model for these data, where x represents the year? A. ƒ1x2 = 1x - 201322 + 32.9 B. g1x2 = 0.131x - 20132 + 32.9 C. h1x2 = 2.52x - 2013 + 32.9 D. k1x2 = -1x - 201324 + 32.9 (a) Make a scatter diagram of the data. (b) Use the regression feature of a calculator to find the best-fitting linear function for the data. Graph the function with the data. (c) Repeat part (b) for a cubic function. (d) Estimate the minimum sight distance for a car traveling 43 mph using the functions from parts (b) and (c). (e) By comparing graphs of the functions in parts (b) and (c) with the data, decide which function best fits the given data. 101. Government Spending on Health Research and Training The table lists the annual amount (in billions of dollars) spent by the federal government on health research and training programs over a 10-yr period. 100. Water Pollution Copper in high doses can be lethal to aquatic life. The table lists copper concentrations in freshwater mussels after 45 days at various distances downstream from an electroplating plant. The concentration C is measured in micrograms of copper per gram of mussel x kilometers downstream. x 5 21 37 53 59 C 20 13 9 6 5 Data from Foster, R., and J. Bates, “Use of mussels to monitor point source industrial discharges,” Environ. Sci. Technol.; Mason, C., Biology of Freshwater Pollution, John Wiley & Sons. (a) Make a scatter diagram of the data. (b) Use the regression feature of a calculator to find the best-fitting quadratic function for the data. Graph the function with the data. (c) Repeat part (b) for a cubic function. (d) By comparing graphs of the functions in parts (b) and (c) with the data, decide which function best fits the given data. (e) Concentrations above 10 are lethal to mussels. Find the values of x (using the cubic function) for which this is the case.
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