Question 1173639
Let's break down how to model and calculate the US federal income tax for a single filer in 2020.

**a) Mathematical Model**

We can represent the tax calculation as a piecewise function, where each piece corresponds to a tax bracket.

Let:

* `I` = Annual Taxable Income
* `T(I)` = Total Tax

The tax brackets and rates are:

1.  **10% Bracket:**
    * If $0 ≤ I ≤ $9,875, then `T(I) = 0.10 * I`
2.  **12% Bracket:**
    * If $9,876 ≤ I ≤ $40,125, then `T(I) = 0.10 * 9,875 + 0.12 * (I - 9,875)`
3.  **22% Bracket:**
    * If $40,126 ≤ I ≤ $85,525, then `T(I) = 0.10 * 9,875 + 0.12 * (40,125 - 9,875) + 0.22 * (I - 40,125)`
4.  **24% Bracket:**
    * If $85,526 ≤ I ≤ $163,300, then `T(I) = 0.10 * 9,875 + 0.12 * (40,125 - 9,875) + 0.22 * (85,525 - 40,125) + 0.24 * (I - 85,525)`
5.  **32% Bracket:**
    * If $163,301 ≤ I ≤ $207,350, then `T(I) = 0.10 * 9,875 + 0.12 * (40,125 - 9,875) + 0.22 * (85,525 - 40,125) + 0.24 * (163,300 - 85,525) + 0.32 * (I - 163,300)`
6.  **35% Bracket:**
    * If $207,351 ≤ I ≤ $518,400, then `T(I) = 0.10 * 9,875 + 0.12 * (40,125 - 9,875) + 0.22 * (85,525 - 40,125) + 0.24 * (163,300 - 85,525) + 0.32 * (207,350 - 163,300) + 0.35 * (I - 207,350)`
7.  **37% Bracket:**
    * If $I ≥ $518,401, then `T(I) = 0.10 * 9,875 + 0.12 * (40,125 - 9,875) + 0.22 * (85,525 - 40,125) + 0.24 * (163,300 - 85,525) + 0.32 * (207,350 - 163,300) + 0.35 * (518,400 - 207,350) + 0.37 * (I - 518,400)`

**b) Calculating Tax for a Monthly Income of $13,750**

1.  **Calculate Annual Income:**
    * Annual Income (I) = $13,750 * 12 = $165,000

2.  **Determine the Applicable Tax Bracket:**
    * $165,000 falls within the $163,301 to $207,350 bracket (32%).

3.  **Calculate the Tax:**
    * T(I) = 0.10 * 9,875 + 0.12 * (40,125 - 9,875) + 0.22 * (85,525 - 40,125) + 0.24 * (163,300 - 85,525) + 0.32 * (165,000 - 163,300)
    * T(I) = 987.5 + 3,630 + 9,988 + 18,666 + 544
    * T(I) = $33,815.5

Therefore, the annual income tax would be $33,815.5.