A firm is trying to create a cost function to represent the costs of producing denim jeans. They figure that the general cost of operating the machine for a whole day is $120, regardless of how many jeans they make that day. In addition to this, they figure that each pair of jeans will cost about $13 to make (for cost of materials, labor,etc). Define the fixed and variable costs, then formulate an appropriate cost function.



To formulate the cost function for the firm producing denim jeans, we first need to identify the fixed and variable costs.

### Fixed Cost:
The fixed cost is the cost that remains constant regardless of the number of jeans produced. In this case, the fixed cost is the general cost of operating the machine for a whole day, which is $120.

### Variable Cost:
The variable cost depends on the number of jeans produced. Here, the cost to make each pair of jeans (materials, labor, etc.) is $13. Therefore, if $ x $ represents the number of jeans produced, the total variable cost will be $ 13x $.

### Cost Function:
The total cost function, $ C(x) $, combines both the fixed and variable costs. It can be expressed as:

\[
C(x) = 120 + 13x
\]

Where:
- $ 120 $ is the fixed cost (in dollars),
- $ 13x $ is the variable cost (in dollars) depending on the number of jeans $ x $ produced.

### Summary:
- **Fixed Cost:** $120
- **Variable Cost:** $13 per pair of jeans
- **Cost Function:** $ C(x) = 120 + 13x $

Would you like details or have any questions?

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Here are five related questions:
1. How would the cost function change if the fixed cost increased to $150 per day?
2. What happens to the total cost if no jeans are produced on a given day?
3. How would the cost function be affected if the cost per pair of jeans decreased to $10?
4. How can you determine the number of jeans that need to be produced to cover a specific cost?
5. What is the break-even point if the selling price per pair of jeans is $20?

**Tip:** Understanding the components of the cost function helps in determining how changes in production levels affect total costs.

A firm is trying to create a cost function to represent the costs of producing denim jeans. They figure that the general cost of operating the machine for a whole day is $120, regardless of how many jeans they make that day. In addition to this, they figure that each pair of jeans will cost about $13 to make (for cost of materials, labor, etc). Define the fixed and variable costs, then formulate an appropriate cost function.

Learn how to formulate a cost function for the production of denim jeans by understanding fixed and variable costs. This problem covers key algebraic concepts, such as linear functions, and provides a step-by-step solution suitable for students in Grades 9-12.

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