Problem 115 - Algebra and Trigonometry

Question

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 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)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

For Part a, assuming the calculated value was say 16.2%, if the value from the graph was 16.4%, then the model underestimates by 0.2%. For Part b, if the calculated value of 'x' was say 4.5, it means that the 18.5% spending would be reached in 4.5 years after 2006, so approximated to 2011.

Step 1: Calculate the percentage spent on healthcare in 2009

Use the function \(f(x)=1.2 \ln x+15.7\) to determine the percentage spent in 2009. Here, 'x' is the number of years after 2006, so for 2009, x = 2009 - 2006 = 3. So, plug in x=3 into the function to find the percentage.

Step 2: Compare with actual value

Compare the computed value to the value from the graph, and determine if the computed value underestimates or overestimates the actual value. Calculate the difference.

Step 3: Predict the year for 18.5% healthcare spending

For part b, use the model to find the year when 18.5% of the GDP is spent on healthcare. To do this, set \(f(x) = 18.5\) and solve for 'x'. The value of 'x' will be the number of years after 2006 when the spending would reach 18.5% of GDP. Round this to the nearest year.

Key Concepts

Logarithmic Functions

Logarithmic functions are a critical part of algebra that allow us to model and analyze phenomena that grow at a rate proportional to their current size. In this exercise, the function given is a logarithmic function:

• \( f(x) = 1.2 \ln x + 15.7 \)

Here, \( \ln \) represents the natural logarithm. It's important to understand that the natural logarithm is the inverse of the exponential function, which makes it particularly useful for modeling data that involves growth or decay processes, like the increase in health care spending relative to GDP over time.
The coefficient 1.2 indicates how strongly the GDP percentage changes with each additional year since 2006. The 15.7 is the constant term, indicating the base percentage of GDP allocated to healthcare initially. Understanding logarithmic functions enables us to predict and analyze future trends over time.

Modeling with Functions

Modeling with functions is a powerful tool in algebra that allows us to represent real-world situations mathematically. It helps in understanding and projecting patterns or behaviors using equations. In this context, we are using a logarithmic function to model how healthcare spending as a percentage of GDP changes over time.

• Given function: \( f(x) = 1.2 \ln x + 15.7 \)

To model the years after 2006, \( x \) represents the number of years since 2006. For instance, for the year 2009, \( x = 2009 - 2006 = 3 \).
By plugging \( x = 3 \) into the function, we can compute the predicted percentage of GDP spent on healthcare for that year. This illustrates how modeling includes setting up equations and variables that appropriately capture the conditions of real-world scenarios. Proper modeling allows us to make educated guesses about future data points.

Problem-Solving with Functions

Problem-solving involves using mathematical tools, such as functions, to calculate and infer results, as demonstrated in this exercise.

• First, find specific GDP percentage for a given year.
• Second, forecast when a certain percentage will be achieved.

For the task of finding the GDP percentage spent on healthcare in 2009, the function is used with \( x = 3 \). This gives a calculated result. Comparing this calculation with actual graph data helps determine if the function overestimates or underestimates reality and by how much.
In predicting when 18.5% of GDP will be spent on healthcare, set the function equal to 18.5 and solve for \( x \):\[1.2 \ln x + 15.7 = 18.5\]Solve the equation for \( x \), and then convert this to a calendar year by adding to 2006. Solving scenarios using functions like this helps bridge abstract algebra concepts and tangible applications.