Alright, let's tackle the questions step by step.
### Part A: Calculating the Amount of Taxes Owed
We are given a taxable income of [tex]$37,000. We need to use the appropriate piecewise function or the marginal tax rate chart to calculate the amount of taxes owed.
Given piecewise function:
\[ f(x) = \begin{cases}
0.10x & \text{if } 0 \leq x \leq 10,275 \\
0.12x - 205.50 & \text{if } 10,276 \leq x \leq 41,175 \\
0.22x - 4,323.00 & \text{if } 41,176 \leq x \leq 89,075 \\
0.24x - 6,104.50 & \text{if } 89,076 \leq x \leq 170,050 \\
0.32x - 19,708.50 & \text{if } 170,051 \leq x \leq 215,950 \\
0.35x - 26,187.00 & \text{if } 215,951 \leq x \leq 539,900 \\
0.37x - 36,985.00 & \text{if } x \geq 539,901
\end{cases} \]
Since $[/tex]37,000[tex]$ falls within the range $[/tex]10,276 \leq x \leq 41,175[tex]$, we use the piecewise function:
\[ f(x) = 0.12x - 205.50 \]
Now, let's calculate the taxes owed:
\[ f(37,000) = 0.12(37,000) - 205.50 \]
\[ f(37,000) = 4,440 - 205.50 \]
\[ f(37,000) = 4,234.50 \]
So, the amount of taxes owed on a taxable income of $[/tex]37,000 is \[tex]$4,234.50.
### Part B: Calculating the Effective Tax Rate
The effective tax rate is calculated by taking the ratio of the taxes owed to the taxable income and then multiplying by 100 to get a percentage.
\[ \text{Effective Tax Rate} = \left( \frac{\text{Taxes Owed}}{\text{Taxable Income}} \right) \times 100 \]
From Part A, we know the taxes owed are \$[/tex]4,234.50 and the taxable income is \[tex]$37,000. Substituting these values, we get:
\[ \text{Effective Tax Rate} = \left( \frac{4,234.50}{37,000} \right) \times 100 \]
\[ \text{Effective Tax Rate} \approx 11.4446\% \]
So, the effective tax rate is approximately 11.44%.
### Part C: Comparing the Piecewise Function to the Marginal Tax Rate Chart
The marginal tax rate chart and the piecewise function essentially represent the same information, but in different formats.
1. Marginal Tax Rate Chart: It lists the tax rates applicable to different income ranges. Each range is associated with a specific percentage rate. For instance, the income range \$[/tex]10,276-\[tex]$41,175 has a marginal tax rate of 12%.
2. Piecewise Function: The piecewise function translates these ranges and rates into a mathematical function that outputs the tax amount directly. It adjusts for any base tax amounts already covered in the lower brackets through the constants in each piecewise function. For example, the function $[/tex]0.12x - 205.50[tex]$ for the range \$[/tex]10,276-\[tex]$41,175 accounts for the taxes owed from the lower tax bracket (\$[/tex]0-\$10,275).
In summary, the piecewise function gives a more immediate calculation of taxes owed by incorporating both the marginal tax rate and adjustments for lower brackets, while the marginal tax rate chart provides the fundamental rates that apply to different ranges of income.
