Problem 115 - College Algebra Essentials

Question

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 \(2006 .\) a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2009 .\) 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 \(18.5 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.

Step-by-Step Solution

Answer

a. By calculating, you will find the predicted percent expenditure on health care in 2009. Compare this with the dataset to determine the deviation.\nb. Solve the equation for \( x \) to find when the expenditure will reach 18.5% of the GDP.}

Step 1: Calculation of Expenditure in 2009

To calculate the expenditure on health care in the year 2009, the value of \( x \) is 3 (2009 - 2006 = 3). Substituting \( x = 3 \) into the function \( f(x) = 1.2 \ln x + 15.7 \), the calculation is \( f(3) = 1.2 \ln 3 + 15.7 \). Use a calculator to compute it.

Step 2: Determine the Actual Deviation

By comparing the obtained value from step 1 to the actual value given on the graph, the deviation can be determined. If the computed value is lesser, then it is an underestimate and if it's higher, it's an overestimate.

Step 3: Calculation of the Year When Healthcare Expenditure Will Reach 18.5%

To predict when the healthcare expenditure will reach 18.5%, the equation is set as \( f(x) = 18.5 \). This results in the equation \( 1.2 \ln x + 15.7 = 18.5 \). Solving for \( x \), first subtract 15.7 from both sides, then divide by 1.2, and finally, take the antilogarithm or the exponential function of both sides.

Key Concepts

Gross Domestic Product

Gross Domestic Product, commonly known as GDP, is a vital economic indicator. It measures the total value of all goods and services produced over a specific time period within a nation's borders. This metric is important because it represents the health of a country's economy.
GDP can be broken down into various sectors like healthcare, technology, and agriculture. When we talk about the percentage of the GDP that goes toward healthcare, we are discussing how much of the nation's economic resources are allocated to healthcare services and products.

• GDP is measured over time, usually quarterly or annually.
• An increasing GDP typically indicates economic growth.
• The portion of GDP spent on healthcare can reflect national priorities and economic conditions.

Understanding GDP helps in recognizing how different sectors, including healthcare, contribute to the economy's overall stability and growth.

Healthcare Expenditure

Healthcare expenditure refers to the amount of resources spent on healthcare services and products. It is a critical aspect of a country's budget because it affects the quality and accessibility of healthcare.
In the context of the exercise, determining the percentage of GDP going toward healthcare allows us to gauge how the healthcare sector is funded relative to the nation's economic output.

• Healthcare expenditure includes hospital services, doctor visits, medications, and preventive care.
• Tracking changes in healthcare expenditure is important for policy-making and budget allocation.
• High healthcare expenditure might indicate efficient healthcare systems but can also suggest rising costs.

Analyzing healthcare expenditure within the GDP context helps understand the economic pressures and benefits associated with healthcare investments.

Mathematical Modeling

Mathematical modeling is a way of using mathematical language and calculations to represent real-world phenomena. In the exercise, the function given: \(f(x) = 1.2 \ln x + 15.7\) is a mathematical model.
This model helps us predict how the GDP percentage dedicated to healthcare changes over time.

• Models simplify complex systems, making it easier to study particular aspects.
• They provide a framework to make predictions and test hypotheses.
• Accuracy of models depends on data inputs and assumptions.

By using mathematical models, we can make informed decisions and predictions based on different variables and data trends.

Exponential Equations

Exponential equations involve variables in the form of exponents. Though the function in the exercise uses a logarithm, understanding exponential equations is crucial, as exponential and logarithmic functions are inverses.
These types of equations are used to describe growth and decay processes, such as population growth or radioactive decay.

• Exponential growth appears in contexts where a quantity grows increasingly rapidly.
• Logarithms are used to undo exponential equations, making calculations more manageable.
• Solving logarithmic equations requires understanding their properties and relationships with exponentials.

The underlying concepts of exponential and logarithmic equations facilitate solving real-world problems involving growth, investment return, and natural phenomena modeling.