Problem 77 - College Algebra

Question

The function \(f(t)=-0.14 t^{2}+0.51 t+31.6\) models the U.S. population ages 65 and older, \(f(t),\) in millions, \(t\) years after \(1990 .\) The function \(g(t)=0.54 t^{2}+12.64 t+107.1\) models the total yearly cost of Medicare, \(g(t),\) in billions of dollars, \(t\) years after \(1990 .\) Graph the function \(\frac{g}{f}\) in a \([0,15,1]\) by \([0,60,1]\) viewing rectangle. What does the shape of the graph indicate about the per capital costs of Medicare for the U.S. population ages 65 and over with increasing time?

Step-by-Step Solution

Answer

The shape of the graph of \( h(t) \) will show whether the per capita costs of Medicare for the U.S. population aged 65 and over is increasing or decreasing over time. This answer can best be obtained by graphing this function with the described criteria.

Step 1: Define the new function

First, we need to create a new function \( h(t)=g(t)/f(t) \), which will represent the per capita cost of Medicare for individuals aged 65 and older. This can be obtained by dividing function \( g(t) \) by function \( f(t) \).

Step 2: Graph the new function

Next, we graph our created function \( h(t) \), using a graphing tool. The graphing tool must have a viewing rectangle of \([0,15,1]\) by \([0,60,1]\). This indicates that the x-axis (time in years) runs from 0 to 15, while the y-axis (cost in billion dollars) runs from 0 to 60.

Step 3: Interpret the graph

Last, we interpret the shape of the generated graph. Look at the trend of the graph over time. If the graph shows an increasing trend, it indicates that the per capita costs of Medicare for the population 65 and over is increasing with time. If the graph trends downward, it indicates decreasing per capita costs.

Key Concepts

Per Capita Cost

Per capita cost is an essential concept in economics and healthcare. It represents the cost per person. In our context, we are calculating the per capita Medicare cost for U.S. citizens aged 65 and over.
This is important because it gives a direct understanding of how healthcare expenses are spread among individuals within a specified group. By modeling this per capita cost using functions, we can identify trends and patterns over time. The function obtained by dividing the total Medicare cost by the population helps us analyze if expenses are increasing or decreasing per individual.

Here, the function \( h(t) = \frac{g(t)}{f(t)} \) allows us to visualize per capita costs graphically. This visualization reveals changes over time, supporting data-driven decision-making for policy adjustments or financial planning.

Medicare Costs

Medicare costs refer to the total expenditure on Medicare services within a year. These costs can fluctuate due to various factors such as policy changes, healthcare inflation, or shifts in population demographics.

The function \( g(t) = 0.54t^2 + 12.64t + 107.1 \) models these costs. With its quadratic form, it offers us insights into how Medicare spending evolves over time. A quadratic model suggests that costs could potentially rise faster than linear models would predict, indicating a possibly increasing financial burden over the years.
Understanding such trends in Medicare costs is crucial for government officials, policymakers, and the financial sector as they prepare budgets and plans for sustainable healthcare funding.

Population Modeling

Population modeling is used to predict how a population will change over time. In our exercise, the function \( f(t) = -0.14t^2 + 0.51t + 31.6 \) models the population aged 65 and older.

The variables in this polynomial function account for growth and fluctuations among this age demographic. The coefficients of the function indicate the rate of change, allowing for predictions on how many will need Medicare services in the future.

• Linear Term (\( 0.51t \)): Reflects an overall steady growth.
• Quadratic Term (\( -0.14t^2 \)): Introduces curvature, hinting at slowing growth over time.

Population predictions assist bureaucrats and health planners in crafting future healthcare infrastructure and resource allocation.

Polynomial Functions

Polynomial functions, which may appear complex, are quite valuable in modeling real-world phenomena. These functions, such as the ones used for Medicare costs and population models in our exercise, involve terms of varying degrees.

A polynomial function has multiple terms like a constant, a linear term, and higher-degree terms like a quadratic term. These are expressed generally as \(ax^n + bx^{n-1} + ... + z\), where \(n\) is the degree.

• Polynomial:
  The sum of multiple terms, each with a variable raised to a non-negative power.
• Quadratic:
  A polynomial of degree 2, with a term \(x^2\).

Understanding polynomial functions is key to interpreting complex models, making predictions, and analyzing the underlying behavior of variables influencing Medicare costs and population data.