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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 \times 0.05 = 2,500 $
- Tax on the remaining $75,000 = $ 75,000 \times 0.12 = 9,000 $

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

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 $ x $.

\[
T(x) = 
\begin{cases} 
0.05x & \text{if } x \leq 50,000 \\
0.05(50,000) + 0.12(x - 50,000) & \text{if } x > 50,000
\end{cases}
\]

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!

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.

Learn how to calculate the tax on an income of $125,000 using a two-part tax policy with 5% for the first $50,000 and 12% for the rest. Understand the step-by-step solution and how to model the tax using a piecewise function.

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