In a particular state and for a particular​ year, the income tax rates for a single person were as follows. A 

33​% 

tax of the first 

​$81008100 

​earned, 

3.53.5​% 

of the next 

​$32003200 

​earned, 

4.254.25​% 

of the next 

​$22502250 

​earned, 

5.95.9​% 

of the next 

​$71007100 

​earned, 

6.456.45​% 

of the next 

​$56 comma 60056,600 

​earned, 

6.656.65​% 

of the next 

​$128 comma 800128,800 

​earned, 

6.856.85​% 

of the next 

​$823 comma 400823,400 

​earned, and 

8.768.76​% 

of any amount earned over 

​$1 comma 029 comma 4501,029,450. 

Let​ f(x) represent the amount of tax owed on an income of x dollars. Answer parts​ (a) through​ (f) below and in parts​ (a) through​ (d) explain in a sentence what the answer means. f(12 comma 00012,000​)equals=enter your response here

This means a tax of 

​$enter your response here 

was due on an income of 

​$enter your response here.

​(Type integers or decimals rounded to the nearest hundredth as needed. Do not include the​ $ symbol in your​ answers.)

To solve this problem, we'll calculate the tax for an income of $ x = 12,000 $ based on the tax rates and income brackets provided.

### Tax brackets and rates:
1. **33%** tax on the first **$8,100**
2. **3.5%** tax on the next **$3,200**
3. **4.25%** tax on the next **$2,250**
4. **5.9%** tax on the next **$7,100**
5. **6.45%** tax on the next **$56,600**
6. **6.65%** tax on the next **$128,800**
7. **6.85%** tax on the next **$823,400**
8. **8.76%** tax on any amount over **$1,029,450**

Since the income is $12,000, only the first two tax brackets will be applied, because $12,000 is less than the sum of the first two brackets ($8,100 + $3,200 = $11,300). We'll calculate the tax as follows:

### Step-by-step calculation:

1. **For the first $8,100**:
   \[
   \text{Tax} = 33\% \times 8,100 = 0.33 \times 8,100 = 2,673
   \]

2. **For the next $3,200**:
   \[
   \text{Tax} = 3.5\% \times 3,200 = 0.035 \times 3,200 = 112
   \]

3. **For the remaining $12,000 - 11,300 = 700** (which is part of the third bracket taxed at 4.25%):
   \[
   \text{Tax} = 4.25\% \times 700 = 0.0425 \times 700 = 29.75
   \]

### Total tax:
\[
\text{Total tax} = 2,673 + 112 + 29.75 = 2,814.75
\]

### Interpretation:

- $ f(12,000) = 2,814.75 $
  
This means a tax of **$2,814.75** was due on an income of **$12,000**.

---

Would you like more details, or do you have any further questions?

Here are 5 related questions to deepen understanding:
1. How would the tax change if the income were $20,000?
2. What is the total tax for an income of $60,000?
3. How does the progressive nature of this tax system affect individuals with different income levels?
4. How would you describe the impact of each tax bracket on overall tax liability?
5. What would be the tax due on an income of $1,000,000?

**Tip:** When dealing with tax brackets, always break down the income into chunks that correspond to each bracket for accurate calculations.

In a particular state and for a particular year, the income tax rates for a single person were as follows. A 33% tax of the first $8,100 earned, 3.5% of the next $3,200 earned, 4.25% of the next $2,250 earned, 5.9% of the next $7,100 earned, 6.45% of the next $56,600 earned, 6.65% of the next $128,800 earned, 6.85% of the next $823,400 earned, and 8.76% of any amount earned over $1,029,450. Let f(x) represent the amount of tax owed on an income of x dollars. Answer parts (a) through (f) below and in parts (a) through (d) explain in a sentence what the answer means. f(12,000) equals=enter your response here. This means a tax of $enter your response here was due on an income of $enter your response here.

Learn how to calculate the income tax on $12,000 based on progressive tax brackets. The problem involves calculating taxes at 33%, 3.5%, and 4.25% rates for different income portions. Explore step-by-step how progressive tax systems work and the final tax amount for this income level. Suitable for Grades 10-12.

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