Question

the given bar graph shows the percentage of gross domestic product (gdp) in a state going toward health care from 2007 through 2010, with projections for 2014 and 2019. the data are modeled by the function f(x) = 1.5 ln x + 12.1, where f(x) is the percentage of gross domestic product going toward health care x years after 2006. use this information to solve parts a and b.\
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• use the function to determine the percentage of gross domestic product that went toward health care in 2009. does this underestimate or overestimate the percent displayed by the graph? by how much?\
• determine the percentage of gross domestic product that went toward health care in 2009 using the given function.\

(do not round until the final answer. then round to the nearest tenth as needed.)\
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bar graph titled percentage of gdp with years 2007 (12.2), 2008 (12.7), 2009 (14.1), 2010 (14.4), 2014 (14.7), 2019 (16.6) on the x - axis and percentage on the y - axis from 0 to 21

Explanation:

Step1: Determine x for 2009

2009 is \(2009 - 2006 = 3\) years after 2006, so \(x = 3\).

Step2: Substitute x into f(x)

The function is \(f(x)=1.5\ln x + 12.1\). Substitute \(x = 3\):
\(f(3)=1.5\ln(3)+12.1\)
Calculate \(\ln(3)\approx1.0986\), then \(1.5\times1.0986\approx1.6479\)
Add 12.1: \(1.6479 + 12.1 = 13.7479\approx13.7\) (rounded to nearest tenth)

Step3: Compare with graph

From the graph, 2009's percentage is 14.1. The calculated value (13.7) is less than 14.1, so it underestimates.
Difference: \(14.1 - 13.7 = 0.4\)

Answer:

The percentage using the function is approximately 13.7%, it underestimates the graph's percentage by 0.4%.