Problem 107 - Precalculus Essentials

Question

Here is the Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer for a recent year. (TABLE CANNOT COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{ccc} 0.10 x & \text { if } & 0379,150 \end{array}\right.$$ Use this information to solve. Find and interpret \(T(20,000)\)

Step-by-Step Solution

Answer

The tax owed, \(T(20,000)\), for an income of \$20,000 is \$2575.

Step 1: Identify the Relevant Tax Bracket

First, we locate the income bracket that \$20,000 falls into. From the given tax rates, \$20,000 fits into the range \$8500 < x ≤ \$34,500, as \$20,000 lies within this range.

Step 2: Apply the Function for The Identified Bracket

The function we have for this tax bracket is \(T(x) = 850.00 + 0.15(x - 8500)\). Here, \$850 represents the tax for the first \$8500 and 0.15(x - 8500) represents the additional 15% tax for the amount that exceeds \$8500.

Step 3: Compute T(20000)

We substitute \(x = 20000\) into our function. This gives us \(T(20000) = 850.00 + 0.15(20000 - 8500) = 850.00 + 0.15*11500\). This simplifies to \(T(20000) = 850.00 + 1725 = \$2575\). Therefore, a taxpayer with a taxable income of \$20,000 would owe \$2575 in tax.

Key Concepts

Understanding Tax Brackets

Tax brackets are a method used by governments to apply different tax rates to different portions of an individual's income. Each bracket corresponds to a specific range of income and comes with its own tax rate. This allows for a progressive tax system where the tax rate increases with the level of income. In the case of our exercise, the Federal Tax Rate Schedule gives different tax rates for income ranges like \(0 < x \leq 8500\), \(8500 < x \leq 34,500\), and so on. The income is divided into "chunks," and each chunk is taxed at the respective rate for that bracket. For example, if you earn \(20,000, part of your income is taxed at the rate for income under \)8,500, and the rest is taxed at the rates for higher brackets.

How Tax Calculation Works

Calculating taxes using tax brackets involves applying specified tax rates to different portions of income. In the exercise, we've determined that \(20,000 falls into the bracket \)8,500 < x \leq 34,500\(.

• The first \)8,500 is taxed at 10%, which amounts to \(850 in taxes.
• The remaining income, which is \)20,000 - \(8,500 = \)11,500, falls in the 15% tax bracket.

The tax for this portion is calculated as \(0.15 \times 11,500 = 1,725\). Adding together the taxes from each bracket, we have a total tax of $2,575. This process ensures that individuals pay taxes proportionate to their earning levels.

Role of Mathematical Modeling

Mathematical modeling allows us to represent real-world situations, like tax computations, using mathematical expressions. In this exercise, we use a piecewise function to model the Federal Tax Rate Schedule. A piecewise function is made up of multiple sub-functions, each corresponding to different parts of the domain—in this case, different income ranges for taxation. Each sub-function models how the tax should be calculated based on where the taxable income lies. For instance, if our income fits into the \(8,500 to \)34,500 range, we use a specific formula \(T(x) = 850.00 + 0.15(x - 8500)\). Mathematical modeling not only helps simplify the computation process but also allows us to efficiently handle different tax systems by accounting for various tax brackets.