document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164633>Question 1164633</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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. \r<BR>\n" );
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document.write( "FEDERAL TAX BRACKETS 2018  -  2019	    Tax rate (%)<BR>\n" );
document.write( "$0  -  $9,525	                                   10<BR>\n" );
document.write( "$9,526  -  $38,700	                           12<BR>\n" );
document.write( "$38,701  -  $82,500	                           22<BR>\n" );
document.write( "$82,501  -  $157,500	                           24<BR>\n" );
document.write( "$157,501  -  $200,000	                          32<BR>\n" );
document.write( "$200,001  -  $500,000	                          35<BR>\n" );
document.write( "$500,001+	                                  37\r<BR>\n" );
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document.write( "Table : 1 Federal income tax brackets\r<BR>\n" );
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document.write( "Required:\r<BR>\n" );
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document.write( "i)	Create a Mathematical Model that represents the aforementioned information to calculate the annual income tax.\r<BR>\n" );
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document.write( "ii)	How much annual income tax you will have to pay if your monthly taxable income is $ 40,000?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854337 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2261)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "brackets = [<BR>\n" );
document.write( "    (0, 9525, 0.10),<BR>\n" );
document.write( "    (9525, 38700, 0.12),<BR>\n" );
document.write( "    (38700, 82500, 0.22),<BR>\n" );
document.write( "    (82500, 157500, 0.24),<BR>\n" );
document.write( "    (157500, 200000, 0.32),<BR>\n" );
document.write( "    (200000, 500000, 0.35),<BR>\n" );
document.write( "    (500000, float('inf'), 0.37)<BR>\n" );
document.write( "]\r<BR>\n" );
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document.write( "cumulative_taxes = []<BR>\n" );
document.write( "prev_limit = 0<BR>\n" );
document.write( "total_tax = 0\r<BR>\n" );
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document.write( "for low, high, rate in brackets[:-1]:<BR>\n" );
document.write( "    tax_in_bracket = (high - low) * rate<BR>\n" );
document.write( "    total_tax += tax_in_bracket<BR>\n" );
document.write( "    cumulative_taxes.append(total_tax)\r<BR>\n" );
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document.write( "print(f\"Cumulative taxes at thresholds: {cumulative_taxes}\")\r<BR>\n" );
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document.write( "def calculate_tax(income):<BR>\n" );
document.write( "    tax = 0<BR>\n" );
document.write( "    for low, high, rate in brackets:<BR>\n" );
document.write( "        if income > low:<BR>\n" );
document.write( "            taxable_amount = min(income, high) - low<BR>\n" );
document.write( "            tax += taxable_amount * rate<BR>\n" );
document.write( "        else:<BR>\n" );
document.write( "            break<BR>\n" );
document.write( "    return tax\r<BR>\n" );
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document.write( "monthly_income = 40000<BR>\n" );
document.write( "annual_income = monthly_income * 12<BR>\n" );
document.write( "tax_due = calculate_tax(annual_income)\r<BR>\n" );
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document.write( "print(f\"Annual Income: {annual_income}\")<BR>\n" );
document.write( "print(f\"Tax Due: {tax_due}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Cumulative taxes at thresholds: [952.5, 4453.5, 14089.5, 32089.5, 45689.5, 150689.5]<BR>\n" );
document.write( "Annual Income: 480000<BR>\n" );
document.write( "Tax Due: 143689.5\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "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.\r<BR>\n" );
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document.write( "### **i) Mathematical Model for Annual Income Tax**\r<BR>\n" );
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document.write( "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:\r<BR>\n" );
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document.write( "* **Bracket 1 (10%):** $9,525 \times 0.10 = \$952.50$<BR>\n" );
document.write( "* **Bracket 2 (12%):** $952.50 + (38,700 - 9,525) \times 0.12 = \$4,453.50$<BR>\n" );
document.write( "* **Bracket 3 (22%):** $4,453.50 + (82,500 - 38,700) \times 0.22 = \$14,089.50$<BR>\n" );
document.write( "* **Bracket 4 (24%):** $14,089.50 + (157,500 - 82,500) \times 0.24 = \$32,089.50$<BR>\n" );
document.write( "* **Bracket 5 (32%):** $32,089.50 + (200,000 - 157,500) \times 0.32 = \$45,689.50$<BR>\n" );
document.write( "* **Bracket 6 (35%):** $45,689.50 + (500,000 - 200,000) \times 0.35 = \$150,689.50$\r<BR>\n" );
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document.write( "**The Piecewise Mathematical Model $T(x)$:**\r<BR>\n" );
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document.write( "$$T(x) = <BR>\n" );
document.write( "\begin{cases} <BR>\n" );
document.write( "0.10x & 0 \le x \le 9,525 \\<BR>\n" );
document.write( "952.50 + 0.12(x - 9,525) & 9,525 < x \le 38,700 \\<BR>\n" );
document.write( "4,453.50 + 0.22(x - 38,700) & 38,700 < x \le 82,500 \\<BR>\n" );
document.write( "14,089.50 + 0.24(x - 82,500) & 82,500 < x \le 157,500 \\<BR>\n" );
document.write( "32,089.50 + 0.32(x - 157,500) & 157,500 < x \le 200,000 \\<BR>\n" );
document.write( "45,689.50 + 0.35(x - 200,000) & 200,000 < x \le 500,000 \\<BR>\n" );
document.write( "150,689.50 + 0.37(x - 500,000) & x > 500,000 <BR>\n" );
document.write( "\end{cases}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **ii) Calculation for Monthly Taxable Income of $40,000**\r<BR>\n" );
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document.write( "**Step 1: Determine Annual Taxable Income**<BR>\n" );
document.write( "$$\text{Annual Income } (x) = \$40,000 \times 12 = \$480,000$$\r<BR>\n" );
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document.write( "**Step 2: Identify the Tax Bracket**<BR>\n" );
document.write( "The income of $\$480,000$ falls into the **35% bracket** ($200,001$ to $500,000$).\r<BR>\n" );
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document.write( "**Step 3: Apply the Model**<BR>\n" );
document.write( "Using the corresponding piece of the function:<BR>\n" );
document.write( "$$T(480,000) = 45,689.50 + 0.35(480,000 - 200,000)$$<BR>\n" );
document.write( "$$T(480,000) = 45,689.50 + 0.35(280,000)$$<BR>\n" );
document.write( "$$T(480,000) = 45,689.50 + 98,000$$<BR>\n" );
document.write( "$$T(480,000) = \$143,689.50$$\r<BR>\n" );
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document.write( "**Final Result:**<BR>\n" );
document.write( "If your monthly taxable income is $\$40,000$, your total annual federal income tax would be **$\$143,689.50$**. </SPAN>\n" );
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