Problem 103 - A Graphical Approach to Precalculus with Limits

Question

The table lists poverty-level income cutoffs for a family of four for selected years. Use the midpoint formula to estimate the poverty-level cutoffs (rounded to the nearest dollar) in 2012 and 2014. $$\begin{aligned} &\\\ &\begin{array}{c|c} \text { Year } & \text { Income (in dollars) } \\ 2011 & 22,350 \\ 2013 & 23.550 \\ 2015 & 24,250 \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

Estimated cutoffs: 2012 - $22,950; 2014 - $23,900.

Step 1: Understanding the Midpoint Formula

The midpoint formula is used to find an average of two values. It is given by \( \frac{x_1 + x_2}{2} \). In this problem, we will use this formula to estimate poverty-level income cutoffs for years between given data points.

Step 2: Estimating the Cutoff for 2012

To estimate the poverty-level cutoff for 2012, we take the midpoint of the income values for 2011 and 2013. Using the formula, this is \( \frac{22,350 + 23,550}{2} \). Calculate the expression to find the income for 2012.

Step 3: Calculating the Midpoint for 2012

Calculate \( \frac{22,350 + 23,550}{2} = \frac{45,900}{2} = 22,950 \). Thus, the estimated poverty-level income cutoff for a family of four in 2012 is $22,950.

Step 4: Estimating the Cutoff for 2014

Next, estimate the poverty-level cutoff for 2014. We take the midpoint of the income values for 2013 and 2015. Using the midpoint formula, this is \( \frac{23,550 + 24,250}{2} \). Calculate this expression to find the income for 2014.

Step 5: Calculating the Midpoint for 2014

Calculate \( \frac{23,550 + 24,250}{2} = \frac{47,800}{2} = 23,900 \). Thus, the estimated poverty-level income cutoff for a family of four in 2014 is $23,900.

Key Concepts

Precalculus

Precalculus is an important mathematical subject that helps prepare students for the study of calculus. One key concept in precalculus is understanding and using the midpoint formula. This formula helps in finding the average value of two numbers, or in other words, the point that is exactly halfway between them. This is particularly useful in a variety of real-world applications like estimating data points. For example, when looking at income levels across different years, it helps predict figures for years not directly observed.
To apply the midpoint formula, take two values, say \(x_1\) and \(x_2\), and compute their average using:

• Midpoint Formula: \( \frac{x_1 + x_2}{2} \)

In our original example, this approach was used to estimate poverty-level incomes between given data points. This technique allows us to make educated guesses about unknown quantities by using data from known points.

Estimation

Estimation is a fundamental skill in mathematics, particularly in situations where exact data is not available or calculations are complex. It involves making reasonable guesses based on known information. For the problem at hand, we make use of the midpoint formula to estimate the poverty-level income for certain years.
Estimations are beneficial in:

• Predicting future trends based on past data.
• Filling gaps in incomplete data sets.
• Making quick calculations when precision is less critical.

For instance, using the midpoint between known poverty incomes for 2011 and 2013, we estimated the income for 2012 to be $22,950. Similarly, the formula helped estimate an income of $23,900 for 2014. These estimations provide insights when direct data is unavailable.

Poverty-Level Income

Poverty-level income represents the minimum amount of earnings needed for individuals or families to meet their basic living expenses. It acts as a guideline for determining who needs financial assistance. Policymakers use these figures to assess economic support systems, allocate resources, and develop strategies to alleviate poverty.
Understanding the trends in poverty-level income over time helps in:

• Tracking economic growth or decline.
• Formulating welfare policies.
• Assessing living standards and inflation impacts.

In our case study, we explored how to estimate the poverty-level income for certain years using midpoint calculations. As incomes fluctuate, it is crucial to use estimation techniques to keep these figures up to date, ensuring they reflect current economic realities.