 firm will break even​ (no profit and no​ loss) as long as revenue just equals cost. The value of x​ (the number of items produced and​ sold) where 

Upper C left parenthesis x right parenthesis equals Upper R left parenthesis x right parenthesisC(x)=R(x) 

is called the​ break-even point. Assume that the below table can be expressed as a linear function.

Find ​(a) the cost​ function, ​(b) the revenue​ function, and ​(c) the profit function.

​(d) Find the​ break-even point and decide whether the product should be​ produced, given the restrictions on sales.

Fixed cost

Variable cost

Price of item

​$300300

​$2020

​$3030

According to the​ restriction, no more than 

2121 

units can be sold.

Question content area bottom

Part 1

​(a) The cost function is 

​C(x)equals=300 plus 20 x300+20x.

​(Simplify your​ answer.)

Part 2

​(​b)

 The revenue function is 

​R(x)equals=30 x30x.

​(Simplify your​ answer.)

Part 3

​(c) The profit function is 

​P(x)equals=enter your response here.

​(Simplify your​ answer.)




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.

The firm will break even (no profit and no loss) as long as revenue just equals cost. The value of x (the number of items produced and sold) where C(x)=R(x) is called the break-even point. Assume that the below table can be expressed as a linear function. Find (a) the cost function, (b) the revenue function, (c) the profit function, and (d) the break-even point and decide whether the product should be produced, given the restrictions on sales. Fixed cost: $300, Variable cost: $20, Price of item: $30, Max units that can be sold: 21.

Learn how to calculate the break-even point with a given cost, revenue, and profit function in a business context. This problem includes a fixed cost of $300, a variable cost of $20 per unit, and a price of $30 per unit. Understand how linear functions model business decisions and why the product should not be produced if the break-even point exceeds the sales limit of 21 units.

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