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Let's go through the problem step by step.

---

### Given:
- Cost to produce each unit: \$140
- Selling price per unit: \$230
- Fixed cost: \$1,200

#### a) **Break-even point:**
The break-even point occurs when total costs equal total revenue.

1. **Revenue function $ R(x) $:**  
   $ R(x) = 230x $, where $ x $ is the number of units sold.
   
2. **Cost function $ C(x) $:**  
   $ C(x) = 140x + 1200 $, where $ 140x $ is the variable cost and \$1,200 is the fixed cost.

3. **Break-even point condition:**  
   $ R(x) = C(x) $

\[
230x = 140x + 1200
\]
Solving for $ x $:

\[
230x - 140x = 1200
\]
\[
90x = 1200
\]
\[
x = \frac{1200}{90} = 13.33
\]

So, the break-even point is at approximately **14 units** (since the number of units must be a whole number).

---

#### b) **Profit function:**
Profit is the difference between revenue and cost.

\[
P(x) = R(x) - C(x)
\]
Substitute the functions:

\[
P(x) = 230x - (140x + 1200)
\]
Simplify:

\[
P(x) = 230x - 140x - 1200
\]
\[
P(x) = 90x - 1200
\]

Thus, the **profit function** is $ P(x) = 90x - 1200 $.

---

#### c) **Profit for 2,000 units:**

Substitute $ x = 2000 $ into the profit function:

\[
P(2000) = 90(2000) - 1200
\]
\[
P(2000) = 180,000 - 1200
\]
\[
P(2000) = 178,800
\]

So, the profit for producing and selling **2,000 units** is **\$178,800**.

---

Would you like further details or clarifications? Here are 5 questions to expand on this:

1. How does the profit change with the increase in the cost per unit?
2. What if the fixed cost increased to \$2,000? How would the break-even point be affected?
3. Can the profit function be used to predict losses? If so, how?
4. What happens if the selling price is reduced by 10%? How does that affect the break-even point?
5. How does the break-even analysis help businesses in pricing their products?

**Tip:** Break-even analysis is a crucial tool in understanding the relationship between cost, revenue, and profit. It helps businesses make informed decisions on production levels and pricing strategies.

For a new product, a manufacturer spends $140 to produce each unit that sells for $230 per unit. There is also a fixed cost of $1,200. a. Find the break-even point. b. What is the profit function for this commodity? c. How much profit is generated when 2,000 units of the product are produced?

Learn how to calculate the break-even point, determine the profit function, and analyze profits for 2,000 units produced in a scenario where a manufacturer spends $140 per unit with a fixed cost of $1,200 and sells the product for $230 each. Suitable for high school students.

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