Problem 91 - College Algebra

Question

Here is the 2011 Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer. (TABLE CAN'T 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: $$T(x)=\left\\{\begin{array}{c}0.10 x \\\850.00+0.15(x-8500) \\\4750.00+0.25(x-34,500) \\\17,025.00+0.28(x-83,600) \\\\\frac{?}{?}\end{array}\right.$$ if \(\quad 0 < x \leq 8500\) if \(\quad 8500 < x \leq 34,500\) if \(\quad 34,500 < x \approx 83,600\) if \(\quad 83,600 < x =174,400\) if \(174,400 < x \leq 379,150\) if \(\quad x >379,150\). Use this information to solve. Find and interpret \(T(20,000)\)

Step-by-Step Solution

Answer

The tax owed by a single taxpayer with a taxable income of $20,000 is $2575.00.

Step 1: Identify Range for the Income

The income of $20,000 falls in the range \$8,500 < x ≤ \$34,500 according to the ranges given in the piecewise function.

Step 2: Apply Corresponding Function

For this income range, the corresponding function in the piecewise function is \(T(x) = \$850.00 + 0.15(x - 8,500)\). We plug $20,000 into this equation as \(x\).

Step 3: Compute the Tax

Using the function, we do the following arithmetic: \(T(20,000) = \$850.00 + 0.15(20,000 - 8,500) = \$850 + 0.15*11,500 = \$850 + 1725 = \$2575.00.\)

Step 4: Interpret the Result

The calculated tax, \$2575.00, is the amount of tax owed by a single taxpayer with a taxable income of $20,000 according to the given tax rate schedule.

Key Concepts

Tax Rate Schedule

Understanding how a tax rate schedule works is crucial for effectively calculating the amount of taxes owed by a taxpayer. In essence, a tax rate schedule is a chart that helps determine the tax rate applicable to an individual's income level. Different ranges of income are taxed at different rates, often increasing progressively as the income level rises. Tax schedules can be complex, with federal, state, and local levels each potentially having their own schedules.

For example, in a typical progressive tax system like that of the United States, a taxpayer with a lower income would fall into a lower tax bracket, hence paying a lower percentage of their income in taxes. Conversely, higher-income earners fall into higher tax brackets, which impose a greater tax rate on portions of their income that exceed certain thresholds.

It's important to note that these rates typically apply to taxable income, which is the portion of income that is subject to tax after adjustments, deductions, and exemptions.

Taxable Income Calculation

The calculation of taxable income is a vital step before utilizing the tax rate schedule. Taxable income is not merely an individual's total earnings but is their gross income minus allowable deductions and exemptions. Allowable deductions can include things like interest on student loans, contributions to retirement savings, or expenses related to self-employment. Exemptions may include personal exemptions or those for dependents.

To calculate taxable income, a taxpayer must start with their gross income and subtract various deductions and exemptions as allowed by tax laws, arriving at an adjusted figure that is subject to income tax. This concept is often illustrated in educational problems with simplified numbers, as shown with the exercise, where a gross income is assumed to equal taxable income. The aforementioned piecewise function is applied to the calculated taxable income to find out the exact tax owed.

For instance, in our exercise, the taxable income calculation was not shown; however, the given income implies that adjustments have been made, and $20,000 is the taxable amount. That amount then locates its appropriate place in the tax rate schedule.

Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions to represent real-world scenarios, making them easier to analyze and understand. Models can take various forms, including equations, functions, graphs, or simulations, and are widely used in a multitude of fields ranging from physics to finance.

In the context of taxation, mathematical modeling helps in expressing complex tax codes as piecewise functions, which break down the tax owed into different parts, each applicable to a specific income range. This visualizes the tax structure in a more manageable way, making it easier to apply to individual cases, as seen in the exercise solution.

Such models not only simplify computation but also assist in predictions and what-if analyses. For instance, with the piecewise function provided, one can easily determine the tax impact of a salary increase or the benefit of a potential deduction, by simply plugging in the new taxable income value into the model.