the given bar graph shows the percentage of gross domestic product (gdp) in a state going toward health care…

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)=1.4lnx + 15.1, 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.\na. 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.\n16.6 %\n(do not round until the final answer. then round to the nearest tenth as needed.)\ndoes 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.\na. overestimates by %\nb. underestimates by 0.3 %\nb. according to the model, when will 18.6% of gross domestic product go toward health care? round to the nearest year.\n(round to the nearest year as needed.)

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)=1.4lnx + 15.1, 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.\na. 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.\n16.6 %\n(do not round until the final answer. then round to the nearest tenth as needed.)\ndoes 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.\na. overestimates by %\nb. underestimates by 0.3 %\nb. according to the model, when will 18.6% of gross domestic product go toward health care? round to the nearest year.\n(round to the nearest year as needed.)

Answer

Explanation:

Step1: Find x - value for 2009

Since x is years after 2006, for 2009, $x = 2009 - 2006=3$.

Step2: Calculate f(x) for x = 3

Substitute $x = 3$ into $f(x)=1.4\ln x + 15.1$. So $f(3)=1.4\ln3+15.1$. Using a calculator, $\ln3\approx1.0986$, then $f(3)=1.4\times1.0986 + 15.1=1.53804+15.1 = 16.63804\approx16.6%$. The graph shows 16.9% for 2009. The difference is $16.9 - 16.6=0.3%$, so it underestimates by 0.3%.

Step3: Solve for x when f(x) = 18.6

Set $1.4\ln x+15.1 = 18.6$. First, subtract 15.1 from both sides: $1.4\ln x=18.6 - 15.1=3.5$. Then divide both sides by 1.4: $\ln x=\frac{3.5}{1.4}=2.5$. Using the property $y = \ln x\Leftrightarrow x = e^{y}$, we have $x = e^{2.5}$. Using a calculator, $e^{2.5}\approx12.18249\approx12$. Since x is years after 2006, the year is $2006 + 12=2018$.

Answer:

a. Underestimates by 0.3% b. 2018