Question 1164633: The U.S. currently has seven federal income tax brackets, with rates of 10%, 12%, 22%, 24%, 32%, 35% and 37%. If you fall into the 37% bracket, that doesn’t mean that the entirety of your taxable income will be subject to a 37% tax. Instead, 37% is your top marginal tax rate.
FEDERAL TAX BRACKETS 2018 - 2019 Tax rate (%)
$0 - $9,525 10
$9,526 - $38,700 12
$38,701 - $82,500 22
$82,501 - $157,500 24
$157,501 - $200,000 32
$200,001 - $500,000 35
$500,001+ 37
Table : 1 Federal income tax brackets
Required:
i) Create a Mathematical Model that represents the aforementioned information to calculate the annual income tax.
ii) How much annual income tax you will have to pay if your monthly taxable income is $ 40,000?
Answer by CPhill(2261)   (Show Source):
You can put this solution on YOUR website! ```python?code_reference&code_event_index=2
brackets = [
(0, 9525, 0.10),
(9525, 38700, 0.12),
(38700, 82500, 0.22),
(82500, 157500, 0.24),
(157500, 200000, 0.32),
(200000, 500000, 0.35),
(500000, float('inf'), 0.37)
]
cumulative_taxes = []
prev_limit = 0
total_tax = 0
for low, high, rate in brackets[:-1]:
tax_in_bracket = (high - low) * rate
total_tax += tax_in_bracket
cumulative_taxes.append(total_tax)
print(f"Cumulative taxes at thresholds: {cumulative_taxes}")
def calculate_tax(income):
tax = 0
for low, high, rate in brackets:
if income > low:
taxable_amount = min(income, high) - low
tax += taxable_amount * rate
else:
break
return tax
monthly_income = 40000
annual_income = monthly_income * 12
tax_due = calculate_tax(annual_income)
print(f"Annual Income: {annual_income}")
print(f"Tax Due: {tax_due}")
```
```text?code_stdout&code_event_index=2
Cumulative taxes at thresholds: [952.5, 4453.5, 14089.5, 32089.5, 45689.5, 150689.5]
Annual Income: 480000
Tax Due: 143689.5
```
To calculate the federal income tax based on the provided progressive tax brackets, we first define the mathematical structure of the system and then apply it to the specific income provided.
### **i) Mathematical Model for Annual Income Tax**
Let $x$ represent the **annual taxable income**. The total annual income tax $T(x)$ is a piecewise function. To build this model, we calculate the cumulative tax paid at the top of each bracket:
* **Bracket 1 (10%):** $9,525 \times 0.10 = \$952.50$
* **Bracket 2 (12%):** $952.50 + (38,700 - 9,525) \times 0.12 = \$4,453.50$
* **Bracket 3 (22%):** $4,453.50 + (82,500 - 38,700) \times 0.22 = \$14,089.50$
* **Bracket 4 (24%):** $14,089.50 + (157,500 - 82,500) \times 0.24 = \$32,089.50$
* **Bracket 5 (32%):** $32,089.50 + (200,000 - 157,500) \times 0.32 = \$45,689.50$
* **Bracket 6 (35%):** $45,689.50 + (500,000 - 200,000) \times 0.35 = \$150,689.50$
**The Piecewise Mathematical Model $T(x)$:**
$$T(x) =
\begin{cases}
0.10x & 0 \le x \le 9,525 \\
952.50 + 0.12(x - 9,525) & 9,525 < x \le 38,700 \\
4,453.50 + 0.22(x - 38,700) & 38,700 < x \le 82,500 \\
14,089.50 + 0.24(x - 82,500) & 82,500 < x \le 157,500 \\
32,089.50 + 0.32(x - 157,500) & 157,500 < x \le 200,000 \\
45,689.50 + 0.35(x - 200,000) & 200,000 < x \le 500,000 \\
150,689.50 + 0.37(x - 500,000) & x > 500,000
\end{cases}$$
---
### **ii) Calculation for Monthly Taxable Income of $40,000**
**Step 1: Determine Annual Taxable Income**
$$\text{Annual Income } (x) = \$40,000 \times 12 = \$480,000$$
**Step 2: Identify the Tax Bracket**
The income of $\$480,000$ falls into the **35% bracket** ($200,001$ to $500,000$).
**Step 3: Apply the Model**
Using the corresponding piece of the function:
$$T(480,000) = 45,689.50 + 0.35(480,000 - 200,000)$$
$$T(480,000) = 45,689.50 + 0.35(280,000)$$
$$T(480,000) = 45,689.50 + 98,000$$
$$T(480,000) = \$143,689.50$$
**Final Result:**
If your monthly taxable income is $\$40,000$, your total annual federal income tax would be **$\$143,689.50$**.
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