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.

a. 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.)
does this underestimate or overestimate the percent displayed by the graph? select the correct choice below and fill in the answer box to complete your choice.
a. overestimates by \\(\square\\)%
b. underestimates by \\(\square\\)%

b. according to the model, when will 16.3% of gross domestic product go toward health care? round to the nearest year.
(round to the nearest year as needed.)

Explanation:

Step1: Determine x for 2009

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

Step2: Calculate f(3)

Substitute \( x = 3 \) into \( f(x) = 1.5\ln x + 12.1 \).
First, find \( \ln 3 \approx 1.0986 \).
Then, \( 1.5 \times 1.0986 + 12.1 \approx 1.6479 + 12.1 = 13.7479 \approx 13.7\% \) (given as 13.7% in the problem, maybe a typo or pre - calculated).
From the graph, 2009 has 14.1%.

Step3: Find the difference

\( 14.1 - 13.7 = 0.4\% \). So the model underestimates by 0.4%.

Step4: Solve for x when f(x)=16.3

Set \( 1.5\ln x + 12.1 = 16.3 \).
Subtract 12.1: \( 1.5\ln x = 16.3 - 12.1 = 4.2 \).
Divide by 1.5: \( \ln x=\frac{4.2}{1.5} = 2.8 \).
Exponentiate: \( x = e^{2.8}\approx 16.44 \).
2006 + 16.44≈2022 (rounded to nearest year).

Answer:

a. Underestimates by 0.4%
b. 2022