Question

survey approximates the number of americans that are age 65 and older and projects that by the year 2050, approximately 83.3 million americans will be at least 65. the bar graph shows the estimated number of americans with projected figures for the year 2020 and beyond. graphing calculator screen displays an exponential function that models the u.s. population age 65 and over, y, in millions, x years after 1899. use this information to solve (a)-(d) below. click the icon to view the bar graph. expreg y = a·b^x a = 3.4508666 b = 1.0230482 explain why an exponential function was used to model the population data. a. an exponential function was used because population is always modeled using exponential functions. b. an exponential function was used because there are too many data points to use a linear function. c. an exponential function was used because exponential functions are always more accurate than linear functions. d. an exponential function was used because the data in the bar graph is increasing more and more rapidly. use the graphing calculator screen to express the model in function notation, with numbers rounded to three decimal places.

Explanation:

Exponential functions are suitable for data showing rapid growth. In population - related data, when the growth rate is not constant but increases over time, an exponential model is appropriate. Here, the bar graph data of the U.S. population age 65 and over is increasing more rapidly, which is a characteristic of exponential growth. For the function notation, we substitute the rounded values of \(a\) and \(b\) into the exponential function form \(y = a\cdot b^{x}\).

Answer:

1. D. An exponential function was used because the data in the bar graph is increasing more and more rapidly.
2. \(y = 3.451\times1.023^{x}\)