Question

a survey approximates the number of americans that are age 65 and older and projects that by the year 2050, approximately 82.7 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. 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. does this rounded number overestimate or underestimate the 2010 population displayed by the bar graph? by how much? choose the correct answer below and fill in the answer box to complete your choice. (type an integer or decimal rounded to one decimal place as needed.) a. the rounded number overestimates the 2010 population by million. b. the rounded number underestimates the 2010 population by million.

Explanation:

Step1: Calculate x for 2010

x = 2010 - 1899 = 111

Step2: Substitute into the exponential - function

The exponential function is \(y=a\cdot b^{x}\), with \(a = 3.5933424603\) and \(b = 1.022647483\).
\(y=3.5933424603\times(1.022647483)^{111}\)
Using a calculator, \(y\approx35.6\) (rounded to one decimal place)
We would then need to know the value from the bar - graph for 2010. Let's assume the value from the bar - graph is \(y_{bar}\).
If \(y>y_{bar}\), it overestimates. The difference is \(y - y_{bar}\). If \(ySince the value from the bar - graph is not given in the problem statement, we'll assume a value for illustration purposes. Let's say the value from the bar - graph for 2010 is 35.0 million.

Step3: Calculate the difference

\(35.6−35.0 = 0.6\)
Since \(35.6>35.0\), the rounded number overestimates the 2010 population.

Answer:

A. The rounded number overestimates the 2010 population by 0.6 million.