Problem 73 - College Algebra and Calculus: An Applied Approach

Question

The per capita personal income in the United States from 1998 to 2005 can be approximated by the linear equation \(y=944.7 t+19,898, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998\. Use the model to estimate the year in which the per capita personal income was \(\$ 32,000\).

Step-by-Step Solution

Answer

Upon calculation, the approximate value of 't' comes to almost 13. Hence the per capita personal income was approximately \$32,000 in the year 2011 (1998 + 13).

Step 1: Set the equation

First, set the provided income value, \$32,000, equal to the given formula \(y=944.7 t+19,898\). This would turn into \(32,000=944.7 t+19,898\)

Step 2: Solve for 't'

Next, solve for 't'. To do this, first subtract \(19,898\) from both sides of the equation which results in \(12,102=944.7 t\). Then, to isolate 't', divide both sides of the equation with \(944.7\). This results in \(t=\frac{12,102}{944.7}\)

Step 3: Calculate 't' and find the corresponding year

Finally, calculate the value of 't' with the obtained equation, and then add 8 to it to find the corresponding year. 't=8' corresponds to the year 1998, so 't= value calculated' will correspond to '1998+ value calculated-8'

Key Concepts

Per Capita Income

Per capita income is a key economic indicator that measures the average income earned per person in a given area in a specified year. It provides an idea of the living standards and economic well-being of a population. In the case of the United States, this metric helps to understand the economic growth and financial health over time.
To calculate per capita income:

• Take the total income of a specific area.
• Divide it by the total population of that area.

With the use of linear equations, we can easily model how per capita income changes over time, offering predictions that can guide economic policies.

United States Economic Data

United States economic data is a rich source of information that encompasses various statistical metrics, including per capita income, GDP, and employment rates. This data is crucial for economists, policy makers, and businesses to make informed decisions.
Specifically, per capita income data allows:

• Comparison of different states or metropolitan areas.
• Analysis of economic trends over time.
• Understanding of income distribution and inequality.

Analyzing this data with tools like linear equations helps to project future trends based on historical data.

Year Estimation

Estimating the year for a specific per capita income involves solving a linear equation that models income over time. This process allows us to predict when a certain income level will be reached according to the given model.
Here's the general approach:

• Use the given linear equation, which links the year variable to per capita income.
• Set up the equation by equating the predicted income to a known value, as seen in the exercise.
• Solve for the time variable to find the estimation of the year.

This technique helps in planning and forecasting, providing a timeline for expected economic outcomes.

College Algebra

College Algebra provides the foundational tools and skills to deal with equations and mathematical models involving unknown variables. It involves algebraic expressions, linear equations, and functions, among others.
In the context of this exercise:

• You apply algebraic manipulation to solve linear equations for a particular variable, such as time.
• Understanding of linear relationships offers insights into how changes in one variable affect another.

Mastery in College Algebra is essential for tackling various mathematical problems, including real-life applications like economic forecasting.

Solving Equations

Solving equations is a critical skill in mathematics that involves finding the values of variables that make an equation true. This process can include:

• Isolating the variable by performing inverse operations.
• Manipulating the equation to simplify complex expressions.

In the exercise, you solve for the variable 't' in a linear equation to estimate a future year when a specific economic condition, like a per capita income level, is met.
Successfully solving equations helps in modeling scenarios and understanding outcomes based on given situations. This skill is incredibly useful not only in academia but also in various practical and professional fields.