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Let's break down the task outlined in the image into its components, then solve the questions step by step:

### **Prompt: Earning Commissions**

A new employee at the company earns:
- **$25,000 base salary** per year.
- **8% commission** on any furniture sold over **$2,000** in sales.

The goal is to complete the table and answer the questions based on this salary and commission structure.

### **1. Completing the Table**

We will calculate the yearly salary based on the sales values given in the table. If sales are $2,000 or less, the salary is just the base salary. For sales over $2,000, the commission applies to the amount beyond $2,000.

| Sales ($) | Yearly salary ($) |  
|-----------|-------------------|  
| 0         | 25,000            |  (Base salary)  
| 1,500     | 25,000            |  (No commission under $2,000)  
| 2,000     | 25,000            |  (No commission at $2,000)  
| 5,000     | 25,000 + 8% of (5,000 - 2,000) = 25,240 |  
| 10,000    | 25,000 + 8% of (10,000 - 2,000) = 25,640 |  
| 50,000    | 25,000 + 8% of (50,000 - 2,000) = 27,840 |  
| 100,000   | 25,000 + 8% of (100,000 - 2,000) = 29,840 |  

### **2. Algebraic model for sales ≤ $2,000**

Since no commission applies for sales ≤ $2,000, the employee's salary is simply the base salary of $25,000. 

So, the algebraic model would be:
\[ S = 25,000 \quad \text{for} \quad x \leq 2,000 \]
Where $ S $ is the annual salary and $ x $ is the total sales.

### **3. Algebraic model for sales > $2,000**

For sales greater than $2,000, the employee receives an 8% commission on the amount over $2,000. Therefore, the model is:

\[ S = 25,000 + 0.08(x - 2,000) \quad \text{for} \quad x > 2,000 \]

Where $ S $ is the annual salary and $ x $ is the total sales.

### **4. Graphical model**

For the graphical model, the salary function consists of two parts:
1. A flat line at $25,000 for sales up to $2,000.
2. A linearly increasing slope after $2,000, representing the 8% commission.

The graph would show:
- A flat line from $ x = 0 $ to $ x = 2,000 $ at $ y = 25,000 $.
- From $ x = 2,000 $, the graph would rise with a slope of 0.08, corresponding to the commission.

### **5. How much in sales is needed to earn $40,000?**

To find how much in sales is required to earn $40,000, we use the equation for $ S $ when $ x > 2,000 $:

\[
40,000 = 25,000 + 0.08(x - 2,000)
\]
Solving for $ x $:

\[
40,000 - 25,000 = 0.08(x - 2,000)
\]
\[
15,000 = 0.08(x - 2,000)
\]
\[
\frac{15,000}{0.08} = x - 2,000
\]
\[
187,500 = x - 2,000
\]
\[
x = 189,500
\]

So, the employee would need to sell **$189,500** in furniture to earn $40,000 in total salary for the year.

---

### Do you need further details on any of these solutions?

Here are 5 related questions you might want to explore:
1. How would the commission change if the percentage increased from 8% to 10%?
2. What would be the impact of lowering the base salary but raising the commission percentage?
3. Can you model the total income for two employees with different commission rates and base salaries?
4. How does this salary plan compare to a straight commission-based pay plan without a base salary?
5. What would be the new required sales to achieve a salary of $50,000?

**Tip**: When solving word problems, it's helpful to define variables and express the relationships algebraically before solving. This clarifies the structure of the problem.

The problem is about calculating yearly salaries based on a base salary and an 8% commission after a certain sales threshold. The user is asked to complete a table, create algebraic models, sketch a graph, and solve for a specific salary.

This problem helps calculate yearly salaries with a base of $25,000 and an 8% commission on sales over $2,000. Solve algebraic models for various sales levels, graph the relationship, and find how much sales are needed to earn $40,000.

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