Problem 92 - 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 \(T(50,000)\).

Step-by-Step Solution

Answer

A single taxpayer with a taxable income of $50,000 owes $8625 in federal tax, according to this model of the tax code.

Step 1: Identify the appropriate section

First, find which part of the function corresponds to 50,000. Looking at the ranges of each condition, \( 34,500 < x \leq 83,600 \) applies for \(x = 50,000\).

Step 2: Substitute into the function

Now, replace \(x\) with 50,000 in the identified function \(4750.00+0.25(x-34,500)\). Hence, the tax will be \(4750.00+0.25(50,000-34,500)\). Calculate the tax.

Step 3: Calculate the tax

Subtract $34,500 from $50,000, which is $15,500. Multiply $15,500 by 0.25 (which represents the 25% tax rate), to get $3875. Then add this to $4750 to get a total tax of $8625.

Step 4: Interpret the result

A single taxpayer with a taxable income of $50,000 owes $8625 in federal tax, according to this model of the tax code.

Key Concepts

Federal Tax Rate

The federal tax rate is a vital concept when considering taxes, as it determines how much tax you owe to the government based on your income. These rates are usually defined by brackets, with different percentages applied to different ranges of income. Each bracket specifies a rate at which income within that range is taxed. For example, in 2011, a single taxpayer might face a tax rate of 10% on income up to $8,500, 15% on income from $8,501 to $34,500, and so forth.
This progressive tax system ensures those who earn more income pay a larger percentage in taxes, contributing proportionately to federal revenues. Understanding which tax rate applies to your income is crucial in assessing how much federal tax you will owe.

Taxable Income

Taxable income is the portion of an individual's gross income that is subject to taxation, Essentially, it is your total income minus any deductions or exemptions that apply.
To determine your taxable income, you take into account all sources of income, such as wages, interest, dividends, and profits from property or assets. Then, eligible deductions, credits, and allowances are deducted. These can include standard deductions, student loan interest deductions, and tax credits.

• For example, if you earned $60,000 in gross income but qualify for $10,000 in deductions, your taxable income would be $50,000.

It's key for taxpayers to accurately calculate their taxable income to prevent over or under-payment of taxes. Understanding this concept is essential for tax planning and compliance.

Single Taxpayer Tax Calculation

Single taxpayer tax calculations can initially seem complex, but breaking them down into steps helps simplify the process. As shown in the example above, calculating a single taxpayer's federal tax involves identifying the correct income bracket and applying the corresponding tax rate.
For someone with a taxable income of \(50,000 using a model reflecting 2011 rates:

• First, determine which tax bracket \)50,000 falls into. The income \(34,500 - \)83,600, is taxed at 25%.
• Then, calculate the tax owed at this level by applying the formula: \( 4750 + 0.25(50,000 - 34,500) \)
• This involves computing the excess income over the lower bracket threshold, multiplying by 25%, and adding the base amount of $4,750.

The final figure represents the total tax liability, providing clarity on what is owed to the IRS, offering a clear path for planning financial responsibilities.

Piecewise Function Interpretation

Piecewise functions are mathematical expressions used to model situations where different conditions apply to different ranges of input values. In the context of federal taxes, each section of a piecewise function corresponds to a different tax bracket. This is essential in tax calculations, as each range of taxable income has a different tax rate applied to it.
Using the example from the federal tax model:

• If your taxable income is $50,000, you locate this within its corresponding condition range, \( 34,500 < x \leq 83,600 \), and use the specific function provided for that range.
• The calculation \( 4750 + 0.25(x - 34,500) \) indicates the tax owed for income falling within this segment.

Interpreting piecewise functions requires selecting the correct segment based on the input, reflecting real-life taxes accurately. This process helps taxpayers understand which calculations apply to their specific income situation and supports accurate tax filing.