Problem 49 - Intermediate Algebra

Question

The function \(P(t)=35.8(1.06)^{t}\) approximates the number of people (in millions) in the United States living in poverty, where \(t\) is the number of years after \(2006 .\) Use the function to complete the table below. Round to the nearest tenth. (Source: U.S. Census Bureau) $$ \begin{array}{|l|l|l|l|l|} \hline \text { Year } & 2006 & 2007 & 2008 & 2009 \\ \hline \begin{array}{l} \text { Number in poverty } \\ \text { (in millions) } \end{array} & & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

In 2006: 35.8 million, 2007: 37.9 million, 2008: 40.2 million, 2009: 42.6 million.

Step 1: Understand the function

The function provided is \( P(t) = 35.8(1.06)^t \), where \( t \) represents the number of years after 2006. This means for each year, you will plug in the respective \( t \) value starting from 0 for 2006.

Step 2: Calculate for year 2006

In 2006, \( t = 0 \). Substitute \( t = 0 \) into the function: \( P(0) = 35.8(1.06)^0 = 35.8 \times 1 = 35.8 \). The number of people in poverty in 2006 is approximately 35.8 million.

Step 3: Calculate for year 2007

In 2007, \( t = 1 \). Substitute \( t = 1 \) into the function: \( P(1) = 35.8(1.06)^1 = 35.8 \times 1.06 = 37.948 \). Rounding to the nearest tenth, the number of people in poverty in 2007 is approximately 37.9 million.

Step 4: Calculate for year 2008

In 2008, \( t = 2 \). Substitute \( t = 2 \) into the function: \( P(2) = 35.8(1.06)^2 = 35.8 \times 1.1236 = 40.23488 \). Rounding to the nearest tenth, the number of people in poverty in 2008 is approximately 40.2 million.

Step 5: Calculate for year 2009

In 2009, \( t = 3 \). Substitute \( t = 3 \) into the function: \( P(3) = 35.8(1.06)^3 = 35.8 \times 1.191016 = 42.648167 \). Rounding to the nearest tenth, the number of people in poverty in 2009 is approximately 42.6 million.

Key Concepts

Poverty Estimation

Poverty estimation is a crucial mathematical exercise that helps policymakers and analysts understand the scope and scale of economic challenges faced by a nation. In this scenario, the estimation revolves around predicting the number of individuals living below the poverty line in the U.S. from 2006 onwards using a mathematical function.
The function provided, \( P(t) = 35.8(1.06)^t \), models this estimate. This exponential function offers a way to forecast future data based on past and present trends, assuming consistent growth over time.

• Exponential Functions: This type of function grows at a consistent percentage rate, which in this case is 6% per year, depicting continuous poverty growth.
• Initial Value: The number 35.8 represents the estimated number of individuals (in millions) experiencing poverty in 2006.

For students, understanding poverty estimation in mathematical terms can bridge abstract algebra with real-world applications, enhancing comprehension and engagement.

U.S. Census Bureau Data

The U.S. Census Bureau is the main source of data about the U.S. population, including socio-economic statistics such as poverty rates. Using data from the census, mathematical models can better reflect real-world conditions and offer insights into socio-economic trends.
The provided function derives its initial data from this organization's research, giving it authority and accuracy. As students apply functions like \( P(t) = 35.8(1.06)^t \), they connect theoretical algebraic concepts with practical, evidence-based projections.

• Data Reliability: Utilizing verified data ensures model accuracy, making estimations more dependable.
• Trends and Patterns: Understanding historical data helps predict future trends, revealing patterns crucial for economic planning.
• Statistical Relevance: Such data aids policymakers in making informed decisions based on logged trends.

Integrating these concepts into math exercises grants learners a dual perspective: mathematical rigor and socio-economic significance.

Function Evaluation

Function evaluation is the process of determining the output of a function for specific inputs. In this exercise, students evaluate the function \( P(t) = 35.8(1.06)^t \) by substituting different values of \( t \) to find the number in poverty for corresponding years.
Each step involves:

• Identifying \( t \), representing the number of years since 2006.
• Using the function to calculate the poverty count, illustrating exponential growth.
• Rounding to the nearest tenth for practical interpretation.

This method gives students a hands-on application of mathematical operations, fostering skills required for more complex real-world analyses. It's important to note how evaluation grants students the power to project and predict within algebraic contexts.

Algebraic Modeling

Algebraic modeling uses equations to describe real-world situations. Here, students model the rise in poverty using the function \( P(t) = 35.8(1.06)^t \).
This model captures critical trends in economic data and allows analysts to make estimates:

• Parameter Identification: Establishes key variables and constants such as growth rate and initial values.
• Trend Projection: Enables predictions based on the function's assumptions, vital for understanding possible future scenarios.
• Real-World Application: Highlights mathematical concepts' relevance, showing how equations can reflect reality.

Algebraic modeling demonstrates how mathematics goes beyond theory, using formulas to capture and analyze socio-economic phenomena effectively. This approach links math studies with actionable insights, pivotal for young learners making sense of their knowledge in meaningful ways.