Math Problem Statement
In a certain country, the amount of income tax T on earnings of x dollars is computed according to the formula: T(x) = { 0, 0 ≤ x ≤ 20,000; 0.1x - 2000, 20,000 < x ≤ 100,000; 0.14x - 6000, x > 100,000 }. Kirk has an income of $10,000. How much will they pay in taxes? Ana has an income of $230,000. How much will they pay in taxes? Dazzline has an income of $90,000. How much will they pay in taxes? A politician proposes increasing the tax rate, but only on those making more than $420,000, to 22%. How much more will Kirk pay in taxes?
Solution
Let's break down the tax-related questions from the image:
The formula for tax is piecewise, depending on the income :
\begin{cases} 0 & \text{if } 0 \leq x \leq 20,000 \\ 0.1x - 2,000 & \text{if } 20,000 < x \leq 100,000 \\ 0.14x - 6,000 & \text{if } x > 100,000 \end{cases}$$ ### 1. **Kirk has an income of $10,000. How much will they pay in taxes?** Kirk's income is $10,000, which falls in the first range: $$0 \leq x \leq 20,000$$. According to the tax formula, $$T(x) = 0$$ in this range. So, Kirk pays **$0** in taxes. ### 2. **Ana has an income of $230,000. How much will they pay in taxes?** Ana's income is $230,000, which falls in the third range: $$x > 100,000$$. The tax formula for this range is: $$T(x) = 0.14x - 6,000$$ Substituting Ana's income $$x = 230,000$$: $$T(230,000) = 0.14(230,000) - 6,000 = 32,200 - 6,000 = 26,200$$ So, Ana pays **$26,200** in taxes. ### 3. **Dazzline has an income of $90,000. How much will they pay in taxes?** Dazzline's income is $90,000, which falls in the second range: $$20,000 < x \leq 100,000$$. The tax formula for this range is: $$T(x) = 0.1x - 2,000$$ Substituting Dazzline's income $$x = 90,000$$: $$T(90,000) = 0.1(90,000) - 2,000 = 9,000 - 2,000 = 7,000$$ So, Dazzline pays **$7,000** in taxes. ### 4. **A politician proposes increasing the tax rate for those making more than $420,000 to 22%. How much more will Kirk pay in taxes?** Since Kirk's income is $10,000, which is below $420,000, this proposed tax increase would not affect Kirk. Therefore, Kirk will pay **$0 more** in taxes. --- Would you like more details on any part of the explanation, or have any other questions? Here are 5 related questions you could consider: 1. What is the total tax paid by someone earning exactly $100,000? 2. How much would Ana pay if the tax rate for incomes above $100,000 were increased to 16% instead of 14%? 3. If Dazzline's income increased to $110,000, how much would they then pay in taxes? 4. How would the tax formula change if the first bracket was taxed at 5% instead of 0%? 5. What would be the tax for someone earning exactly $20,000? **Tip:** When dealing with piecewise functions, carefully identify which range the input falls into before applying the formula.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Formulas
T(x) = 0 if 0 ≤ x ≤ 20,000
T(x) = 0.1x - 2000 if 20,000 < x ≤ 100,000
T(x) = 0.14x - 6000 if x > 100,000
Theorems
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Suitable Grade Level
Grades 10-12
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