In a particular state and for a particular​ year, the income tax rates for a single person were as follows. A 4​% tax of the first ​$8300 ​earned, 4.75​% of the next ​$3100 ​earned, 5.25​% of the next ​$2250 ​earned, 5.9​% of the next ​$7100 ​earned, 6.45​% of the next ​$56 comma 500 ​earned, 6.65​% of the next ​$128 comma 600 ​earned, 6.85​% of the next ​$823 comma 400 ​earned, and 8.82​% of any amount earned over ​$1 comma 029 comma 250. 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(50 comma 000​)equals
  
This means a tax of ​$
  
 was due on an income of ​$
  


Let's compute the total tax owed on an income of $50,000 using the tax rates provided. The tax rates are structured progressively, so we will calculate the tax owed for each income range step-by-step until we reach $50,000.

### Tax brackets:
1. **4% on the first $8,300:**  
   \[ \text{Tax} = 0.04 \times 8,300 = 332 \]

2. **4.75% on the next $3,100 (income between $8,301 and $11,400):**  
   \[ \text{Tax} = 0.0475 \times 3,100 = 147.25 \]

3. **5.25% on the next $2,250 (income between $11,401 and $13,650):**  
   \[ \text{Tax} = 0.0525 \times 2,250 = 118.13 \]

4. **5.9% on the next $7,100 (income between $13,651 and $20,750):**  
   \[ \text{Tax} = 0.059 \times 7,100 = 419.90 \]

5. **6.45% on the next $29,250 (income between $20,751 and $50,000):**  
   Since the income falls within this bracket and reaches $50,000, we calculate the tax for the remaining amount:
   \[ 50,000 - 20,750 = 29,250 \]
   \[ \text{Tax} = 0.0645 \times 29,250 = 1,886.63 \]

### Total tax calculation:
Now, we sum up the taxes from all brackets to find the total tax due on $50,000:

\[
\text{Total tax} = 332 + 147.25 + 118.13 + 419.90 + 1,886.63 = 2,903.91
\]

### Final Answer:
The total tax owed on an income of $50,000 is **$2,903.91**.

#### Explanation:
This means a tax of **$2,903.91** was due on an income of **$50,000**.

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Would you like more details or have any further questions? Here are five follow-up questions:

1. How would the tax change if the income were $60,000?
2. How is the total tax affected by increasing each income bracket by 10%?
3. What is the tax rate applied to an income of $120,000?
4. How would the total tax compare if the person were filing jointly with another individual?
5. What would be the marginal tax rate for an income of $200,000?

**Tip:** When calculating taxes in a progressive tax system, always apply each rate to the corresponding income range, not the entire income!

In a particular state and for a particular year, the income tax rates for a single person were as follows. A 4% tax of the first $8300 earned, 4.75% of the next $3100 earned, 5.25% of the next $2250 earned, 5.9% of the next $7100 earned, 6.45% of the next $56,500 earned, 6.65% of the next $128,600 earned, 6.85% of the next $823,400 earned, and 8.82% of any amount earned over $1,029,250. 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(50,000)= This means a tax of $ was due on an income of $

Learn how to calculate the income tax on a $50,000 income using progressive tax rates. This problem involves applying piecewise functions and basic arithmetic to determine the total tax owed. Suitable for students in Grades 9-12.

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