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Let's break down the mathematical content from the image and solve each part step by step.

The scenario involves a small business making and selling cookies at a farmer's market.

### Given Information:
- **Fixed costs** (e.g., for kitchen, rent, etc.): $790 per month.
- **Variable cost per cookie**: $0.24.
- **Selling price per dozen cookies**: $6.00.

Now, I'll go through each question one by one:

### (a) Write a linear cost function representing the cost $ C(x) $ to produce $ x $ dozen cookies per month.
For each dozen cookies:
- There are 12 cookies in a dozen.
- The variable cost per dozen is $ 0.24 \times 12 = 2.88 $.

So, the total cost function combines the fixed cost and the variable cost:
\[
C(x) = 2.88x + 790
\]
where $ x $ is the number of dozen cookies.

### (b) Write a linear revenue function representing the revenue $ R(x) $ for selling $ x $ dozen cookies.
- The revenue from selling one dozen cookies is $6.00.
- So, the revenue function is:
\[
R(x) = 6x
\]
where $ x $ is the number of dozen cookies sold.

### (c) Write a linear profit function representing the profit $ P(x) $ for producing and selling $ x $ dozen cookies in a month.
The profit function is the difference between the revenue and the cost:
\[
P(x) = R(x) - C(x) = 6x - (2.88x + 790)
\]
Simplifying this:
\[
P(x) = 6x - 2.88x - 790 = 3.12x - 790
\]

### (d) Determine the number of cookies (in dozens) that must be produced and sold for a monthly profit.
To find when the business breaks even, we set $ P(x) = 0 $ (no profit, no loss) and solve for $ x $:
\[
3.12x - 790 = 0
\]
\[
3.12x = 790
\]
\[
x = \frac{790}{3.12} \approx 253.21
\]
So, the business must sell approximately **254 dozen cookies** to make a profit.

### (e) If 150 dozen cookies are sold in a given month, how much money will the business make or lose?
We can use the profit function $ P(x) = 3.12x - 790 $ and substitute $ x = 150 $:
\[
P(150) = 3.12(150) - 790 = 468 - 790 = -322
\]
The business would **lose $322** if they sold 150 dozen cookies in a month.

---

Let me know if you need further clarification on any of the steps or more details.

### Relative Questions:
1. How would the cost function change if the variable cost per cookie increased?
2. What would the profit be if the fixed costs were reduced by $100?
3. How would the break-even point change if the selling price per dozen increased to $7?
4. If 300 dozen cookies were sold, what would the total profit be?
5. How does the break-even point shift if both the fixed costs and the selling price per dozen increased?

### Tip:
Always double-check the assumptions (e.g., dozen cookies, costs) when setting up cost and revenue functions to avoid errors!

A small business makes cookies and sells them at the farmer's market. The fixed monthly cost for use of a Health Department-approved kitchen and rental space at the farmer's market is $790. The cost of labor, taxes, and ingredients for the cookies amounts to $0.24 per cookie and the cookies sell for $6.00 per dozen. (a) Write a linear cost function representing the cost C(x) to produce x dozen cookies per month. (b) Write a linear revenue function representing the revenue R(x) for selling x dozen cookies. (c) Write a linear profit function representing the profit P(x) for producing and selling x dozen cookies in a month. (d) Determine the number of cookies (in dozens) that must be produced and sold for a monthly profit. (e) If 150 dozen cookies are sold in a given month, how much money will the business make or lose?

Learn how to write and analyze linear cost, revenue, and profit functions for a small cookie business. This problem covers core concepts in algebra, including cost and revenue calculations and profit determination. Suitable for students in Grades 9-12.

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