Problem 66 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The distribution of income in a certain city can be described by the mathematical model \(y=\left(2.8 \cdot 10^{11}\right)(x)^{-1.5}\), where \(y\) is the number of families with an income of \(x\) or more dollars. a. How many families in this city have an income of \(\$ 20,000\) or more? b. How many families have an income of \(\$ 40,000\) or more? c. How many families have an income of \(\$ 100,000\) or more?

Step-by-Step Solution

Answer

\(y \approx 35000\) #tag_title# (Step 3: Calculate the number of families with an income of \(40,000 or more.)#tag_content# Substitute the given income \(x = 40000\): \(y = (2.8 \cdot 10^{11})(40000)^{-1.5}\) Solve for y: \(y \approx 10000\) #tag_title# (Step 4: Calculate the number of families with an income of \(100,000 or more.)#tag_content# Substitute the given income \(x = 100000\): \(y = (2.8 \cdot 10^{11})(100000)^{-1.5}\) Solve for y: \(y \approx 1000\) #tag_title# (Step 5: Write the final answer.)#tag_content# In this city, there are approximately: a. 35,000 families with an income of \(20,000 or more. b. 10,000 families with an income of \(40,000 or more. c. 1,000 families with an income of \(100,000 or more.

Step 1: (Step 1: Write the given function.)

The income distribution model is given by the function: \(y = (2.8 \cdot 10^{11})(x)^{-1.5}\) Now we need to find the number of families with incomes of \(20,000 or more, \)40,000 or more, and $100,000 or more. We will do this by substituting the corresponding income values for x in the given function.

Step 2: (Step 2: Calculate the number of families with an income of \(20,000 or more.)

Substitute the given income \)x = 20000$: \(y = (2.8 \cdot 10^{11})(20000)^{-1.5}\) Solve for y:

Key Concepts

Mathematical Models

Mathematical models are powerful tools that help us understand complex systems and phenomena in the real world by using mathematical frameworks. They translate real-world scenarios into mathematical language, allowing us to study variables, relationships, and predict outcomes. In the context of income distribution, a mathematical model can describe how income is distributed among families in a particular area, like in the given function for a city's income distribution.The given model for income distribution is expressed as:\[ y = (2.8 \times 10^{11})(x)^{-1.5} \]Where:

• \( y \) is the number of families with an income \( x \) or more dollars.
• \( x \) represents different income levels.

This function involves an inverse power of income, highlighting the model's use of power-law distribution, frequently encountered in fields like economics and social sciences. By substituting specific income values for \( x \), we can estimate the number of families earning above these levels. This model gives us a tool to quantify and analyze how wealth is distributed within a city, and how many family units fall into various income brackets.

Income Inequality

Income inequality refers to the uneven distribution of income within a population. It indicates the gap between the wealthiest and the poorest individuals or families in a community. Mathematically modeling this phenomenon helps us visualize and quantify the extent of inequality, as seen in the income distribution model from the exercise. Studying income inequality is important due to its implications:

• It influences economic growth and stability.
• It impacts social cohesion and equality of opportunity.
• It is crucial for designing effective economic and social policies.

The given function illustrates how many families fall into different income levels, offering insights into the broader income distribution picture. As income inequality persists, understanding its structure can guide authorities in addressing issues like poverty and ensuring more equitable resource distribution. Using mathematical approaches to study income inequality aids in identifying patterns and shifts in distribution, helping inform policy-making and societal improvement strategies.

Applied Mathematics

Applied mathematics uses mathematical techniques and principles to solve practical problems across different fields, from engineering to economics and beyond. In the context of the exercise, applied mathematics is instrumental in forming models like the income distribution function, which helps solve real-world issues by interpreting data patterns and making informed predictions. The process involves:

• Gathering real-world data to identify trends and variables.
• Creating and refining mathematical models to explain these observations.
• Applying these models to make predictions or decisions.

For example, by substituting income values into our model, we can determine how many families fall into specific income ranges. This can guide local policy makers in understanding economic landscapes, making it a staple tool in economic planning and other applied sciences. Hence, applied mathematics bridges theoretical mathematics and practical applications, enabling us to translate complex data into actionable insights.