In a particular state and for a particular​ year, the income tax rates for a single person were as follows. A 3​% tax of the first ​$8300 ​earned, 3.75​% of the next ​$3100 ​earned, 4.5​% of the next ​$2150 ​earned, 5.9​% of the next ​$7200 ​earned, 6.45​% of the next ​$56 comma 600 ​earned, 6.65​% of the next ​$128 comma 800 ​earned, 6.85​% of the next ​$823 comma 100 ​earned, and 8.78​% 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.
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Part 1
​(a)​ f(10 comma 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.)

Let's calculate the tax owed for an income of $x = 10,000$ dollars based on the given tax brackets.

The tax brackets are:

1. 3% on the first $8,300.
2. 3.75% on the next $3,100.
3. 4.5% on the next $2,150.
4. 5.9% on the next $7,200.
5. 6.45% on the next $56,600.
6. 6.65% on the next $128,800.
7. 6.85% on the next $823,100.
8. 8.78% on any amount earned over $1,029,250.

Since $10,000 is in the second tax bracket, we will only need to calculate the tax owed on the first two brackets.

### Step-by-step calculation:
1. **First bracket: 3% on the first $8,300**
   \[
   0.03 \times 8,300 = 249
   \]
   Tax for the first bracket is $249.

2. **Second bracket: 3.75% on the next $1,700 (because $10,000 - $8,300 = $1,700)**
   \[
   0.0375 \times 1,700 = 63.75
   \]
   Tax for the second bracket is $63.75.

### Total tax:
\[
\text{Total tax} = 249 + 63.75 = 312.75
\]

Thus, $ f(10,000) = 312.75 $.

### Interpretation:
This means a tax of **$312.75** was due on an income of **$10,000**.

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Do you have any questions or would you like more details on the solution?

Here are 5 relative questions that expand on this:

1. What would the tax owed be for an income of $15,000?
2. How does the tax change if the income increases by $1,000 in the second bracket?
3. What is the total tax owed for an income of $100,000?
4. How do the tax rates change after an income exceeds $200,000?
5. Can you graph the tax owed as a function of income?

**Tip:** When calculating taxes using brackets, break the income into portions matching the brackets, and compute each portion separately to ensure accuracy.

In a particular state and for a particular year, the income tax rates for a single person were as follows. A 3% tax of the first $8300 earned, 3.75% of the next $3100 earned, 4.5% of the next $2150 earned, 5.9% of the next $7200 earned, 6.45% of the next $56,600 earned, 6.65% of the next $128,800 earned, 6.85% of the next $823,100 earned, and 8.78% 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. Part 1 (a) f(10,000) equals This means a tax of was due on an income of .

Learn how to calculate the income tax owed for an income of $10,000 based on state tax brackets. This problem involves using percentage-based tax rates for different income ranges and combining the results for an accurate total. Suitable for Grades 9-12.

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