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Let's break down the problem step-by-step.

The manufacturer has:
- A fixed monthly cost of **$52,500**.
- A variable production cost of **$8** per unit.
- The product sells for **$13** per unit.

We are asked to find:

1. **Cost function $ C(x) $**, where $ x $ represents the number of units produced.
   \[
   C(x) = \text{Fixed Cost} + (\text{Variable Cost per Unit} \times x)
   \]
   Therefore, 
   \[
   C(x) = 52500 + 8x
   \]

2. **Revenue function $ R(x) $**, where $ x $ is the number of units sold. Since each unit sells for $13:
   \[
   R(x) = 13x
   \]

3. **Profit function $ P(x) $**, which is the revenue minus the cost:
   \[
   P(x) = R(x) - C(x) = 13x - (52500 + 8x)
   \]
   Simplifying,
   \[
   P(x) = 5x - 52500
   \]

Now, to compute the profits for production levels of **8,000** and **13,000** units:

4. **Profit at 8,000 units $ P(8000) $**:
   \[
   P(8000) = 5(8000) - 52500 = 40000 - 52500 = -12500
   \]

5. **Profit at 13,000 units $ P(13000) $**:
   \[
   P(13000) = 5(13000) - 52500 = 65000 - 52500 = 12500
   \]

### Final answers:
- (a) $ C(x) = 52500 + 8x $
- (b) $ R(x) = 13x $
- (c) $ P(x) = 5x - 52500 $
- (d) Profit at 8,000 units: $ P(8000) = -12500 $, Profit at 13,000 units: $ P(13000) = 12500 $

Would you like more details or have any questions?

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**Relative questions:**
1. What would the break-even production quantity be for this manufacturer?
2. How does the profit change as the production quantity increases?
3. What would the profit be if 10,000 units were produced?
4. How does the fixed cost affect the overall cost function?
5. What is the impact of changing the selling price on the revenue function?

**Tip:** The break-even point occurs where the profit function equals zero.

A manufacturer has a monthly fixed cost of $52,500 and a production cost of $8 for each unit produced. The product sells for $13/unit. Find the following functions (in dollars) and compute each profit.

(a) What is the cost function? C(x) = 
(b) What is the revenue function? R(x) = 
(c) What is the profit function? P(x) = 
(d) Compute the profit (loss) corresponding to production levels of 8,000 and 13,000 units.

Learn how to calculate the cost, revenue, and profit functions for a manufacturer with a fixed monthly cost of $52,500, a production cost of $8/unit, and a selling price of $13/unit. Understand how to compute profits for production levels of 8,000 and 13,000 units. Suitable for Grades 9-12.

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