178 CHAPTER 3 Linear and Quadratic Functions temperature (in degrees Fahrenheit) and the number of chirps per second for the striped ground cricket. Temperature (ºF), x Chirps per Second, C 88.6 93.3 80.6 69.7 69.4 79.6 80.6 76.3 75.2 20.0 19.8 17.1 14.7 15.4 15.0 16.0 14.4 15.5 Source: Pierce, George W. The Songs of Insects. Cambridge, MA, Harvard University Press, 1949, pp. 12–21. (a) Using a graphing utility, draw a scatter plot of the data, treating temperature as the independent variable. What type of relation appears to exist between temperature and chirps per second? (b) Based on your response to part (a), find either a linear or a quadratic model that best describes the relation between temperature and chirps per second. (c) Use your model to predict the chirps per second if the temperature is 80 F. ° 2 1. Mixed Practice Which Model? A study was completed to measure the reaction time of a person using the nondominant hand. The data for men in each age range and the corresponding average reaction times are provided in the table. Age Range (Class) Median Age of Each Class, x Average Reaction Time (in milliseconds) 20–29 24.5 483 30–39 34.5 469 40–49 44.5 524 50–59 54.5 559 60–69 64.5 602 Source: National Library of Medicine. (a) Using a graphing utility, draw a scatter plot of the data, treating the median age as the independent variable. What type of relation appears to exist? (b) Based on your response to part (a), find either a linear or a quadratic model. (c) Use the model you found in part (b) to predict the reaction time for a person in the range of 70–79 years of age. 22. Challenge Problem Simpson’s Rule The figure shows the graph of y ax bx c. 2 = + + Suppose that the points h y y , , 0, , 0 1 ( ) ( ) − and h y , 2 ( ) are on the graph. It can be shown that the area enclosed by the parabola, the x-axis, and the lines x h = − and x h = is h ah c Area 3 2 6 2 ( ) = + 18. Height of a Ball A shot-putter throws a ball at an inclination of 45° to the horizontal. The following data represent the height of the ball h, in feet, at the instant that it has traveled x feet horizontally. Distance, x Height, h 20 25 40 40 60 55 80 65 100 71 120 77 140 77 160 75 180 71 200 64 (a) Use a graphing utility to 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 quadratic function of best fit that models the relation between distance and height. (c) Use the function found in part (b) to determine how far the ball will travel before it reaches its maximum height. (d) Use the function found in part (b) to find the maximum height of the ball. (e) With a graphing utility, graph the quadratic function of best fit on the scatter plot. 19. Mixed Practice Which Model? The numbers of annual Netflix subscribers are provided in the table. The variable x represents the year (15 2015, 16 2016, = = and so on), and y is the number of subscribers (in millions). Year Subscribers (in millions) 15 62.71 16 79.9 17 99.04 18 124.35 19 151.56 20 192.95 21 209 22 220.6 (a) Using a graphing utility, draw a scatter plot of the data, treating year as the independent variable. What type of relation appears to exist? (b) Based on your response to part (a), find either a linear or a quadratic model. (c) If this trend continues, predict the number of subscribers for the year 2025. 20. Mixed Practice Which Model? A cricket makes a chirping noise by sliding its wings together rapidly. Perhaps you have noticed that the number of chirps seems to increase with the temperature. The following data list the
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