Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

A certain country taxes the first \(\$ 20,000\) of an individual's income at a rate of \(15 \%\), and all income over \(\$ 20,000\) is taxed at \(20 \%\). Find a piecewise-defined function \(T\) that specifies the total tax on an income of \(x\) dollars.

Step-by-Step Solution

Answer

The tax function is \( T(x) = 0.15x \) if \( x \leq 20,000 \); \( T(x) = 3000 + 0.20(x - 20,000) \) if \( x > 20,000 \).

Step 1: Define the Tax Bracket Limits

Identify the income thresholds at which the tax rates change. In this scenario, the first tax bracket includes incomes up to \( \$20,000 \), and the tax rate changes for any income above that.

Step 2: Calculate Tax for the First Bracket

For income \( x \leq \$20,000 \), the tax rate is \( 15\% \). Thus, the tax for this portion of income is calculated as: \[ T(x) = 0.15x \]

Step 3: Calculate Tax for the Second Bracket

For income \( x > \\(20,000 \), the tax on the first \( \\)20,000 \) is fixed at \( 0.15 \times 20,000 = \\(3,000 \). The portion of income over \( \\)20,000 \) is taxed at \( 20\% \), so the tax on this portion is: \[ 0.20(x - 20,000) \] Therefore, the total tax is: \[ T(x) = 3000 + 0.20(x - 20,000) \]

Step 4: Define the Piecewise Function

Combine the equations from Steps 2 and 3 into one piecewise function for \( T(x) \): \[ T(x) = \begin{cases} 0.15x & \text{if } 0 \leq x \leq 20,000 \ 3000 + 0.20(x - 20,000) & \text{if } x > 20,000 \end{cases} \]

Key Concepts

Tax Calculation

When dealing with tax calculation, it's important to understand how taxes are determined based on different portions of income. Tax systems utilize different rates applied to distinct ranges of income. These are often designed to be fair and progressive, meaning higher income is taxed at higher rates. In the given problem, the income is split into segments, each with its designated tax rate:

• The first segment involves taxing the initial $20,000 at a rate of 15%.
• Any income over this amount is taxed at a higher rate of 20%.

Understanding tax calculation involves being able to apply different mathematical operations to different sections of income to find the total tax amount. Let's break down how these calculations are performed using the given piecewise function.

Income Brackets

Income brackets serve as the foundation of progressive tax systems like the one in our example. Essentially, an income bracket is a range of incomes that are taxed at a specific rate. The purpose of these brackets is to ensure that everyone pays their fair share according to their ability to pay.

For instance, in this example:

• The first income bracket is from $0 to $20,000, taxable at 15%.
• Any income above $20,000 falls into another bracket taxed at 20%.

By setting these income brackets, individuals with different earnings are taxed according to the portion of their income that actually falls into each bracket, rather than being taxed at a flat rate on their entire income. Understanding how these brackets operate helps clarify how the total tax amount is calculated.

Mathematical Modeling

Mathematical modeling is a powerful tool that helps us represent real-world problems, like tax calculation, with mathematical expressions. In this scenario, a piecewise function models how tax is computed for different income levels. The steps outlined in the solution guided us through setting up the function for this purpose.

The piecewise function defined is: \[ T(x) = \begin{cases} 0.15x & \text{if } 0 \leq x \leq 20,000 \ 3,000 + 0.20(x - 20,000) & \text{if } x > 20,000 \end{cases} \]
This function:

• Represents how each part of the income is taxed based on its position within the specified income brackets.
• Shows the change in taxation once the income crosses a certain threshold.

Using mathematical functions, such as piecewise functions, allows us to easily compute the total tax for any income level efficiently, showing the versatility of mathematics in practical applications.