Question 1173946
Absolutely! Let's break down this tax calculation problem.

**a) Mathematical Model**

We'll create a piecewise function to represent the tax brackets. Let `I` be the annual taxable income, and `T(I)` be the total tax owed.

* **10% Bracket:**
    * If 0 ≤ I ≤ $9,875, then T(I) = 0.10 * I
* **12% Bracket:**
    * If $9,876 ≤ I ≤ $40,125, then T(I) = $987.50 + 0.12 * (I - $9,875)
* **22% Bracket:**
    * If $40,126 ≤ I ≤ $85,525, then T(I) = $3,617.50 + 0.22 * (I - $40,125)
* **24% Bracket:**
    * If $85,526 ≤ I ≤ $163,300, then T(I) = $13,627.50 + 0.24 * (I - $85,525)
* **32% Bracket:**
    * If $163,301 ≤ I ≤ $207,350, then T(I) = $32,208.50 + 0.32 * (I - $163,300)
* **35% Bracket:**
    * If $207,351 ≤ I ≤ $518,400, then T(I) = $46,304.50 + 0.35 * (I - $207,350)
* **37% Bracket:**
    * If I ≥ $518,401, then T(I) = $155,142.00 + 0.37 * (I - $518,400)

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

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

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

3.  **Calculate the Tax:**
    * T($165,000) = $32,208.50 + 0.32 * ($165,000 - $163,300)
    * T($165,000) = $32,208.50 + 0.32 * $1,700
    * T($165,000) = $32,208.50 + $544
    * T($165,000) = $32,752.50

**Therefore, the annual income tax would be $32,752.50.**