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(Federal Income Taxes and Piecewise Functions MC)

The piecewise function represents the amount of taxes owed, [tex]f(x)[/tex], as a function of the taxable income, [tex]x[/tex]. Use the marginal tax rate chart or the piecewise function to answer the question.

Marginal Tax Rate Chart
\begin{tabular}{|c|c|}
\hline
Tax Bracket & Marginal Tax Rate \\
\hline
[tex]$\$[/tex]0-\[tex]$10,275$[/tex] & [tex]$10\%$[/tex] \\
\hline
[tex]$\$[/tex]10,276-\[tex]$41,175$[/tex] & [tex]$12\%$[/tex] \\
\hline
[tex]$\$[/tex]41,176-\[tex]$89,075$[/tex] & [tex]$22\%$[/tex] \\
\hline
[tex]$\$[/tex]89,076-\[tex]$170,050$[/tex] & [tex]$24\%$[/tex] \\
\hline
[tex]$\$[/tex]170,051-\[tex]$215,950$[/tex] & [tex]$32\%$[/tex] \\
\hline
[tex]$\$[/tex]215,951-\[tex]$539,900$[/tex] & [tex]$35\%$[/tex] \\
\hline
[tex]$\ \textgreater \ \$[/tex]539,901[tex]$ & $[/tex]37\%[tex]$ \\
\hline
\end{tabular}

\[
f(x) = \begin{cases}
0.10x, & 0 \leq x \leq 10,275 \\
0.12x - 205.50, & 10,276 \leq x \leq 41,175 \\
0.22x - 4,323.00, & 41,176 \leq x \leq 89,075 \\
0.24x - 6,104.50, & 89,076 \leq x \leq 170,050 \\
0.32x - 19,708.50, & 170,051 \leq x \leq 215,950 \\
0.35x - 26,187.00, & 215,951 \leq x \leq 539,900 \\
0.37x - 36,985.00, & x \geq 539,901
\end{cases}
\]

Determine the effective tax rate for a taxable income of $[/tex]\[tex]$95,600$[/tex]. Round the final answer to the nearest hundredth.

A. [tex]$17.00\%$[/tex]
B. [tex]$17.61\%$[/tex]
C. [tex]$22.70\%$[/tex]
D. [tex]$24.00\%$[/tex]



Answer :

GinnyAnswer
To determine the effective tax rate for a taxable income of \[tex]$95,600 using the provided piecewise function, follow these steps: 1. Identify the appropriate tax bracket: Given that the taxable income is \$[/tex]95,600, it falls within the range \[tex]$89,076 to \$[/tex]170,050. According to the piecewise function, the corresponding formula to calculate the tax owed in this range is:
[tex]\[ f(x) = 0.24x - 6,104.50 \][/tex]

2. Calculate the tax owed:
Substitute \[tex]$95,600 as \( x \) in the formula: \[ f(95,600) = 0.24 \cdot 95,600 - 6104.50 \] Compute: \[ 0.24 \cdot 95,600 = 22,944 \] Subtracting the constant: \[ 22,944 - 6,104.50 = 16,839.50 \] Therefore, the tax owed for a taxable income of \$[/tex]95,600 is \[tex]$16,839.50. 3. Determine the effective tax rate: The effective tax rate is defined as the percentage of the taxable income that is paid as taxes. It is calculated using the formula: \[ \text{Effective Tax Rate} = \left(\frac{\text{Tax Owed}}{\text{Taxable Income}}\right) \times 100 \] Substitute the values: \[ \left(\frac{16,839.50}{95,600}\right) \times 100 \] Compute the division: \[ \frac{16,839.50}{95,600} \approx 0.1761 \] Convert the result to a percentage: \[ 0.1761 \times 100 = 17.61\% \] 4. Round the final answer to the nearest hundredth: The calculated effective tax rate is 17.61%. Thus, the effective tax rate for a taxable income of \$[/tex]95,600 is [tex]\(\boxed{17.61\%}\)[/tex].

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