Questions: The table shows the populations y (in millions) of the United States for 2011 through 2020. The variable t represents the years, with t=11 corresponding to 2011. t 11 12 13 14 15 16 17 18 19 20 y 312.0 314.4 316.6 319.0 321.4 323.8 326.2 328.6 331.0 333.4 (a) Plot the data and connect adjacent points with a line segment. Use the slope of each line segment to determine the year when the population increased least rapidly. - 2011 to 2012 - 2012 to 2013 - 2013 to 2014 - 2014 to 2015 - 2015 to 2016 - 2016 to 2017 - 2017 to 2018 - 2018 to 2019 - 2019 to 2020 (b) Find the average rate of change (in millions per year) of the population of the United States from 2011 through 2020. (Round your answer to two decimal places.)

Solution Steps

The slope of each line segment is calculated as the change in population divided by the change in time. For each year interval, the slope is given by:

\[ \text{slope} = \frac{y_{t+1} - y_t}{t+1 - t} \]

Calculating the slopes for each interval:

• 2011 to 2012: \(\frac{314.4 - 312.0}{12 - 11} = 2.4\)
• 2012 to 2013: \(\frac{316.6 - 314.4}{13 - 12} = 2.2\)
• 2013 to 2014: \(\frac{319.0 - 316.6}{14 - 13} = 2.4\)
• 2014 to 2015: \(\frac{321.4 - 319.0}{15 - 14} = 2.4\)
• 2015 to 2016: \(\frac{323.8 - 321.4}{16 - 15} = 2.4\)
• 2016 to 2017: \(\frac{326.2 - 323.8}{17 - 16} = 2.4\)
• 2017 to 2018: \(\frac{328.6 - 326.2}{18 - 17} = 2.4\)
• 2018 to 2019: \(\frac{331.0 - 328.6}{19 - 18} = 2.4\)
• 2019 to 2020: \(\frac{333.4 - 331.0}{20 - 19} = 2.4\)

The year with the least rapid population increase is the one with the smallest slope. From the calculations, the smallest slope is \(2.2\) for the interval 2012 to 2013.

The average rate of change is calculated as the total change in population divided by the total change in time:

\[ \text{Average rate of change} = \frac{333.4 - 312.0}{20 - 11} = \frac{21.4}{9} \approx 2.3778 \]

Rounded to two decimal places, the average rate of change is \(2.38\) million per year.

Final Answer