Math Problem Statement

A certain tax policy states that the first $50,000 of income is taxed at 5% and any income above $50,000 is taxed at 12%. Calculate the tax on $125,000. Write a function that models the tax policy.

Solution

Let's break down the question and solve it step by step.

Problem:

  • The tax policy divides income into two parts:
    • The first $50,000 is taxed at 5%.
    • Any income above $50,000 is taxed at 12%.

We need to:

  1. Calculate the tax on $125,000.
  2. Write a function that models the tax policy.

Part a) Calculating the tax on $125,000:

To calculate the tax on $125,000, we will split the income into two portions:

  • The first $50,000 taxed at 5%.

  • The remaining $125,000 - $50,000 = $75,000 taxed at 12%.

  • Tax on the first $50,000 = 50,000×0.05=2,50050,000 \times 0.05 = 2,500

  • Tax on the remaining $75,000 = 75,000×0.12=9,00075,000 \times 0.12 = 9,000

So, the total tax on $125,000 is: 2,500+9,000=11,5002,500 + 9,000 = 11,500

Thus, the total tax is $11,500.

Part b) Writing a function to model the tax policy:

We can define a piecewise function to model the tax policy based on income xx.

\begin{cases} 0.05x & \text{if } x \leq 50,000 \\ 0.05(50,000) + 0.12(x - 50,000) & \text{if } x > 50,000 \end{cases}$$ Simplified: - For $$x \leq 50,000$$, the tax is $$0.05x$$. - For $$x > 50,000$$, the tax is $$2,500 + 0.12(x - 50,000)$$. This function covers both cases of the tax policy. --- Would you like further explanation or details on these solutions? Here are 5 related questions for more practice: 1. How would the tax change if the income was $60,000 instead of $125,000? 2. What would the tax be for an income of $200,000? 3. How does the piecewise function behave for values very close to $50,000? 4. If the tax rates change to 6% for the first $50,000 and 15% for the rest, how would that affect the tax function? 5. Can you find the total tax for an income of $75,000? **Tip:** Piecewise functions are useful for modeling scenarios where different rules apply to different ranges of values, such as tax brackets!

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Math Problem Analysis

Mathematical Concepts

Tax calculation
Piecewise functions
Percentage

Formulas

Tax on income = percentage × income
Piecewise function for tax calculation

Theorems

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Suitable Grade Level

Grades 9-11

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