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!