Problem 21 - Applied Calculus

Question

The percentage of people, \(P\), below the poverty level in the \(\mathrm{US}^{14}\) is given in Table \(1.4 .\) (a) Find a formula for the percentage in poverty as a linear function of time in years since 2000 . (b) Use the formula to predict the percentage in poverty in 2006 . (c) What is the difference between the prediction and the actual percentage, \(12.3 \%\) ? $$ \begin{array}{l|c|c|c|c} \hline \text { Year (since 2000) } & 0 & 1 & 2 & 3 \\ \hline P \text { (percentage) } & 11.3 & 11.7 & 12.1 & 12.5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The prediction for 2006 is 13.7%, which is 1.4% higher than the actual 12.3%.

Step 1: Understanding the Problem

We need to model the percentage of people below the poverty line using a linear function based on the given data and use this model to predict the percentage in 2006. Finally, we will compare this prediction to the actual value.

Step 2: Identifying Points for Slope Calculation

We have data points \((0, 11.3)\) and \((3, 12.5)\). This will allow us to calculate the slope of the linear function representing the percentage in poverty.

Step 3: Calculating the Slope

The slope \(m\) of the linear function can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the data points, we get \(m = \frac{12.5 - 11.3}{3 - 0} = \frac{1.2}{3} = 0.4\).

Step 4: Formulating the Linear Equation

The linear function can be represented as \(P = mt + b\), where \(m\) is the slope and \(b\) is the y-intercept. Using the initial value \((0, 11.3)\), we find \(b = 11.3\). Thus, the equation is \(P = 0.4t + 11.3\).

Step 5: Predicting the Value in 2006

In 2006, \(t = 6\). Substituting this into our equation: \(P = 0.4 \cdot 6 + 11.3 = 2.4 + 11.3 = 13.7\). Hence, the predicted percentage is 13.7%.

Step 6: Calculating the Difference from Actual Value

The actual percentage in 2006 is 12.3%. The difference between the predicted and actual percentage is \(13.7 - 12.3 = 1.4\).

Step 7: Conclusion of Results

The prediction slightly overestimates the actual value by 1.4%.

Key Concepts

Slope Calculation

The concept of calculating the slope is crucial when working with linear functions. The slope tells us how steep the line is and the direction it moves. For a linear function, the slope is the ratio of the change in the vertical direction (change in percentage of poverty, for example) to the change in the horizontal direction (change in years since 2000 in this case).
To calculate the slope, use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Given data points

• (0, 11.3) representing the year 2000 and a poverty level of 11.3%,
• (3, 12.5) representing the year 2003 and a poverty level of 12.5%.

Plug these into the formula to get \( m = \frac{12.5 - 11.3}{3 - 0} = \frac{1.2}{3} = 0.4 \).
This means that for each year after 2000, the poverty rate increases by 0.4% on average.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In context, it represents the poverty level at the base year, which is 2000 in this scenario. The y-intercept is essentially the starting value of your linear equation.
In a linear equation of the form \( P = mt + b \), where \( P \) is the dependent variable (poverty percentage), \( m \) is the slope, \( t \) is time (years since 2000), and \( b \) is the y-intercept, you determine \( b \) using the initial condition.
From the data point \((0, 11.3)\), it is clear that when \( t = 0 \), \( P = 11.3 \). Therefore, \( b = 11.3 \). This indicates that the percentage of poverty was 11.3% at the start of the observed period.

Data Modeling

Data modeling with linear functions involves creating a mathematical equation that best represents the data points you've collected. It helps in understanding and forecasting trends by producing a simple, yet effective, model.
For our data, we've created the equation \( P = 0.4t + 11.3 \). This model acts as a forecasting tool to determine future values by inputting the year (t) since 2000.
Data modeling with linear functions is useful because:

• It provides a straightforward approach to observe trends.
• It makes predicting future values easier if the trend remains consistent.

In this linear model, small changes in time (years) produce predictable changes in poverty percentage, allowing us to make forecasts such as poverty rates in future years.

Poverty Rate Prediction

Predicting poverty rates with a linear model like \( P = 0.4t + 11.3 \) involves plugging in future values of \( t \) to forecast the rate. For example, for the year 2006, which is 6 years since 2000, the equation becomes:

• \( P = 0.4 \cdot 6 + 11.3 = 2.4 + 11.3 = 13.7 \)%

This method provides a simple way to anticipate poverty trends based on past data.
However, linear predictions have their limitations. They assume a constant rate of change which may not always reflect real-life scenarios as trends can evolve due to socio-economic shifts.
Despite its simplicity, this model is a practical tool for initial assessments, assisting policymakers and researchers in planning and decision-making.