Question

a survey approximates the number of americans that are age 65 and older and projects that by the year 2050, approximately 82.8 million americans will be at least 65. it shows the estimated number of americans with projected figures for the year 2020 and beyond. a 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. a. 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 the data in the bar graph is increasing more and more rapidly. c. an exponential function was used because exponential functions are always more accurate than linear functions. d. an exponential function was used because there are too many data points to use a linear function. b. use the graphing calculator screen to express the model in function notation, with numbers rounded to three decimal places. f(x) =

Explanation:

a. Exponential functions are used when data shows a rapid increase over time. In population - related data, if the growth rate is accelerating, an exponential model is appropriate. Option B correctly states that the data in the graph is increasing more and more rapidly, which is a key characteristic for using an exponential function.
b. The general form of an exponential function is $y = a\cdot b^{x}$. From the calculator screen, $a = 3.5$ and $b=1.02$. So the function in function - notation is $f(x)=3.500\times1.020^{x}$.

Answer:

a. B. An exponential function was used because the data in the graph is increasing more and more rapidly
b. $f(x)=3.500\times1.020^{x}$