212 CHAPTER 4 Polynomial and Rational Functions In Problems 23–30, analyze each polynomial function f by following Steps 1 through 8 on page 210. 23. f x x x x 0.2 1.5876 0.31752 3 2 ( ) = + − − 24. f x x x x 0.8 4.6656 3.73248 3 2 ( ) = − − + 25. f x x x x 2.56 3.31 0.89 3 2 ( ) = + − + 26. f x x x x 2.91 7.668 3.8151 3 2 ( ) = − − − 27. f x x x 2.5 0.5625 4 2 ( ) = − + 28. f x x x 18.5 50.2619 4 2 ( ) = − + 29. f x x x x 2 5 4 4 3 π ( ) = − + − 30. f x x x x 1.2 0.5 3 2 4 2 ( ) = − + − + In Problems 31–42, analyze each polynomial function f by following Steps 1 through 8 on page 207. 31. f x x x 4 3 ( ) = − 32. f x x x3 ( ) = − 33. f x x x x 12 3 2 ( ) = + − 34. f x x x x 2 8 3 2 ( ) = + − 35. f x x x x x 2 12 8 48 4 3 2 ( ) = + − − 36. f x x x x 4 10 4 10 3 2 ( ) = + − − 37. f x x x x x 5 4 3 2 ( ) = − − + + 38. f x x x x x 5 4 20 5 4 3 2 ( ) = − + + − 39. f x x x x 15 80 80 5 4 3 ( ) = + + 40. f x x x x x 3 6 12 24 6 5 4 3 ( ) = + − − 41. f x x x x 1 5 4 5 5 20 3 2 ( ) = − − + 42. f x x x x 3 2 15 4 6 15 3 2 ( ) = − − + Applications and Extensions 43. Hurricanes In 2012, Hurricane Sandy struck the East Coast of the United States, killing 147 people and causing an estimated $75 billion in damage. With a gale diameter of about 1000 miles, Sandy was the largest hurricane ever to form over the Atlantic Basin. Through 2022, Hurricane Wilma is the strongest Atlantic hurricane ever recorded, reaching an intensity of 882 mbar in October 2005. The data in the table show the number of major hurricanes in the Atlantic Basin (category 3, 4, or 5) during each decade from 1921 to 2020. Major Hurricanes Striking Atlantic Basin, H Decade, x 1921–1930, 1 1931–1940, 2 1941–1950, 3 1951–1960, 4 1961–1970, 5 1971–1980, 6 1981–1990, 7 1991–2000, 8 2001–2010, 9 2011–2020, 10 17 16 29 33 27 16 16 27 33 31 Source: National Oceanic & Atmospheric Administration. (a) With a graphing utility draw a scatter plot of the data. Comment on the type of relation that may exist between the two variables. (b) Use a graphing utility to find the cubic function of best fit that models the relation between decade and number of major hurricanes. (c) Use the model found in part (b) to predict the number of major hurricanes that struck the Atlantic Basin between 1961 and 1970. (d) With a graphing utility, graph the cubic function of best fit on the scatter plot. (e) Concern has been raised about the increase in the number and intensity of hurricanes, but some scientists believe the increase is just a natural fluctuation that could last another decade or two. Use your model to predict the number of major hurricanes that will strike the Atlantic Basin between 2021 and 2030. Is your result reasonable? What does this result suggest about the reliability of using a model to predict an event outside the domain of the data? 44. Poverty Rates The data in the table below show the percentage of people in the United States who are below the poverty level. (a) With a graphing utility, draw a scatter plot of the data. Comment on the type of relation that appears to exist between the two variables. (b) Decide on a function of best fit to these data (linear, quadratic, or cubic), and use this function to predict the percent of people in the United States who were below the poverty level in 2022 t 15 . ( ) = Compare your prediction to the actual value of 14.6. (c) Draw the function of best fit on the scatter plot drawn in part (a). Year, t Percent below Poverty Level, P 2008, 1 12.3 2009, 2 12.5 2010, 3 13.2 2011, 4 14.3 2012, 5 15.1 2013, 6 15 2014, 7 14.8 2015, 8 14.8 2016, 9 13.5 2017, 10 12.7 2018, 11 12.3 2019, 12 11.8 2020, 13 10.5 2021, 14 11.6 2022, 15 14.6 Source: worldbank.org
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