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By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)f(x)=1.2 \ln x+15.7\( where \)f(x)\( is the percentage of the U.S. gross domestic product going toward health care \)x\( years after \)2006 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)2008 .\( Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \)18.6 \%$ of the U.S. gross domestic product go toward health care? Round to the nearest year.

Step by step solution

01

Part a: Determine the Percentage for 2008

Plug 2 into the function to represent the year 2008 (this is because x represents years after 2006). This gives us \(f(2) = 1.2 \ln 2 + 15.7\). Using a calculator, compute the value and round to the nearest tenth of a percent.

02

Part a: Compare Findings with Graph

This step requires comparing the value computed in the previous step with the percentage shown in the graph for the year 2008. As we don't have the graph available, this step can't be completed.

03

Part b: Determine the Year when Percentage will be 18.6%

We want to find 'x' when \(f(x) = 18.6\). Setting up the equation gives us \(18.6 = 1.2 \ln x + 15.7\). We need to solve for 'x'. To do that, subtract 15.7 from both sides, then divide by 1.2 and finally take the exponential of both sides of the equation to solve for x. Don't forget to add the base year 2006 to the answer, because x represents years after 2006. Round the answer to the nearest year.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gross Domestic Product

The Gross Domestic Product (GDP) is an essential economic measure. It represents the total monetary value of all goods and services produced over a specific time frame within a nation. To put it simply, GDP is like a scorekeeper for the economy. The higher the GDP, the better the economy is doing.

• When we talk about GDP, we're often interested in the portion of it that relates to different sectors, such as health care, manufacturing, or technology.
• For example, if a large portion of the GDP is spent on health care, it might indicate a focus on public health, potentially higher health care costs, or an aging population.

Understanding GDP helps economists, policymakers, and the public gauge how well an economy is functioning. It can also signal if economic policies need adjustments. GDP is not only vital for comparisons between countries but also within sectors to track changes over time.

Health Care Expenditure

Reflecting the significant role of health care in the modern economy, health care expenditures represent the total value of resources spent on health services and products. These include hospital services, medical research, and medication.

• Tracking these expenditures as a percentage of GDP helps give insight into the emphasis a nation places on healthcare relative to its entire economy.
• In the example problem, projecting future expenditures helps in budget planning and policy formation.
• It can also indicate trends such as increasing health care costs, which may influence health insurance and access to services.

By evaluating this expenditure, stakeholders can make informed decisions regarding funding allocations or necessary policy changes. Seeing health care as a portion of GDP allows stakeholders to assess its efficiency and reach towards economic goals.

Mathematical Modeling

Mathematical modeling employs mathematical techniques to represent real-world scenarios. It allows us to understand and predict complex phenomena.

• In contexts like health care spending, models can project future trends from current data.
• The function \( f(x) = 1.2 \ln x + 15.7 \) models the percentage of GDP allocated to health care, which adjusts over time.
• The use of logarithmic functions helps accommodate for gradual changes and growth which align with real-world data trends.

Models like this provide a scientific basis for decision-making and policy creation in economic contexts. By predicting future outcomes, they help prepare for resource allocation and strategic planning to adapt to potential changes in health expenditure trends.