Let's go step by step and fill in the missing parts based on the given information.

Given:

• Fixed cost = $300
• Variable cost per item = $20
• Price of item = $30
• Maximum units that can be sold = 21 units

Part 1: Cost Function

The cost function $C(x)$ is the total cost of producing $x$ items, which includes both the fixed cost and the variable cost.

$C(x) = \text{Fixed Cost} + (\text{Variable Cost} \times x)$ $C(x) = 300 + 20x$

So, the cost function is: $C(x) = 300 + 20x$

Part 2: Revenue Function

The revenue function $R(x)$ represents the total revenue obtained from selling $x$ items, which is the price per item times the number of items sold.

$R(x) = \text{Price per item} \times x$ $R(x) = 30x$

So, the revenue function is: $R(x) = 30x$

Part 3: Profit Function

The profit function $P(x)$ is the difference between the revenue and the cost:

$P(x) = R(x) - C(x)$ Substituting the values for $R(x)$ and $C(x)$:

$P(x) = 30x - (300 + 20x)$ Simplifying:

$P(x) = 30x - 300 - 20x$ $P(x) = 10x - 300$

So, the profit function is: $P(x) = 10x - 300$

Part 4: Break-even Point

The break-even point occurs when revenue equals cost, i.e., $R(x) = C(x)$.

$30x = 300 + 20x$ Solving for $x$:

$30x - 20x = 300$ $10x = 300$ $x = \frac{300}{10} = 30$

Thus, the break-even point is at 30 units. However, since no more than 21 units can be sold according to the restrictions, the product should not be produced as it is impossible to reach the break-even point.

Let me know if you need further details or have any questions!

Here are five related questions to consider:

1. What is the significance of the break-even point in business analysis?
2. How does changing the price per item affect the revenue and profit functions?
3. If the fixed costs were reduced, how would it impact the break-even point?
4. How would increasing the maximum number of units sold to 40 affect the decision to produce the product?
5. What happens if the variable cost per item increases?

Tip: Always ensure that the break-even point is achievable given production or sales constraints before starting a project.