https://file.aiquickdraw.com/mathbot/file/1726524407947el73xvf5.jpg

Let's break down and solve both of the problems shown in the image:

### Problem 7: Write a piecewise function to model the tiered tax system.
The problem describes a tax system where:
1. Earnings up to **$100,000** are taxed at **35%**.
2. Earnings between **$100,000** and **$500,000** are taxed at **45%**.
3. Earnings above **$500,000** are taxed at **55%**.

We need to write this as a piecewise function that models the tax rate based on the income, denoted as $ E $. The piecewise function can be written as:

\[
T(E) = 
\begin{cases} 
0.35E & \text{if } 0 \leq E \leq 100,000 \\
0.35(100,000) + 0.45(E - 100,000) & \text{if } 100,000 < E \leq 500,000 \\
0.35(100,000) + 0.45(400,000) + 0.55(E - 500,000) & \text{if } E > 500,000 
\end{cases}
\]

Where:
- In the first range, $ T(E) = 0.35E $ means 35% of the earnings up to $100,000.
- In the second range, we first tax the $100,000 at 35% and then tax the amount over $100,000 at 45%.
- In the third range, we first tax $100,000 at 35%, then the next $400,000 at 45%, and finally any amount over $500,000 at 55%.

### Problem 8: Find the value of $ k $ that makes the function continuous.

This question refers to a function (not fully shown in the image) where we are likely trying to adjust a constant $ k $ to make the transition between two pieces of the function smooth, without jumps or breaks.

To solve this, we would need to see the specific function to set the conditions for continuity.

Would you like more details on how to solve the first problem, or can you provide more information for the second one?

### Related Questions:
1. How would the tax model change if the tax brackets were expanded or the rates changed?
2. Can you calculate specific taxes for various income levels using the piecewise function?
3. What is the mathematical concept of continuity and why is it important for smooth functions?
4. How can we verify that the piecewise tax function is correctly applied at the boundaries between tax brackets?
5. How would we graph the tax rate as a function of earnings?

**Tip:** For piecewise functions, always verify the boundaries of each piece to ensure the function transitions smoothly from one segment to the next.

Many income tax systems are calculated using a tiered method. Under a certain tax law, the first $100,000 of earnings are subject to a 35% tax; earnings greater than $100,000 and up to $500,000 are subject to a 45% tax. Any earnings greater than $500,000 are taxed at 55%. Write a piecewise function that models this situation.

Understand how to model a tiered income tax system using piecewise functions. The problem covers earnings taxed at 35%, 45%, and 55%, based on income levels. Perfect for students studying piecewise functions in Grades 11-12.

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