Suppose you are the manager of a firm. he accounting department has provided cost estimates. and the sales department sales estimates, on a new product. Analyze the data they give you, determine what it
will take to break even, and decide whether to go ahead with production of the new product. The product has a production cbst function C(x) = 500x + 18,000 and a revenue function R(x) = 625x
The break-even quantity is how many units?
If the company can produce and sell no more than 262 units should it do so?

To determine the break-even quantity and whether the company should produce the new product, let's analyze the given functions:

### 1. **Cost Function**:  
\[ C(x) = 500x + 18,000 \]  
Where:
- $ C(x) $ is the total cost.
- $ x $ is the number of units produced.
- $ 500x $ represents the variable cost (cost per unit).
- $ 18,000 $ is the fixed cost (constant regardless of units produced).

### 2. **Revenue Function**:  
\[ R(x) = 625x \]  
Where:
- $ R(x) $ is the total revenue.
- $ x $ is the number of units sold.
- $ 625x $ is the revenue per unit sold.

### 3. **Break-even Point**:  
The break-even point occurs when total revenue equals total cost.  
So, set $ R(x) = C(x) $:

\[
625x = 500x + 18,000
\]

### 4. **Solving for $ x $**:
To find the break-even quantity, solve the equation:

\[
625x - 500x = 18,000
\]
\[
125x = 18,000
\]
\[
x = \frac{18,000}{125} = 144 \text{ units}
\]

### Break-even Quantity:  
The firm will break even when it produces and sells **144 units**.

### 5. **Decision on Producing 262 Units**:
Now, if the company can produce and sell up to 262 units, should they do so?  
To decide this, we need to see if producing and selling 262 units generates profit. Compare the cost and revenue for $ x = 262 $.

#### Cost at $ x = 262 $:
\[
C(262) = 500(262) + 18,000 = 131,000 + 18,000 = 149,000
\]

#### Revenue at $ x = 262 $:
\[
R(262) = 625(262) = 163,750
\]

#### Profit at $ x = 262 $:
\[
\text{Profit} = R(262) - C(262) = 163,750 - 149,000 = 14,750
\]

Since the company makes a profit of **$14,750** by producing and selling 262 units, it would be beneficial to go ahead with production.

### Conclusion:
- The break-even quantity is 144 units.
- The company should produce and sell up to 262 units, as it results in a profit of $14,750.

---

Do you want more details on any part of the solution? Here are 5 related questions you might want to explore:

1. What happens if the production cost increases slightly? How does that affect the break-even point?
2. What if the sales price per unit is reduced to $600? What is the new break-even quantity?
3. Can we calculate the maximum profit the company can make with a limited number of units sold?
4. How would the analysis change if the fixed costs were reduced by 20%?
5. If production exceeds 262 units, how do economies of scale impact the cost function?

**Tip:** Always calculate the break-even point before starting any production to avoid losses and ensure profitability.

Suppose you are the manager of a firm. The accounting department has provided cost estimates, and the sales department sales estimates, on a new product. Analyze the data they give you, determine what it will take to break even, and decide whether to go ahead with production of the new product. The product has a production cost function C(x) = 500x + 18,000 and a revenue function R(x) = 625x. The break-even quantity is how many units? If the company can produce and sell no more than 262 units, should it do so?

Analyze the break-even point for a new product with a cost function C(x) = 500x + 18,000 and a revenue function R(x) = 625x. Find out if the company should produce up to 262 units and explore detailed profit calculations. Ideal for Grades 9-12 students interested in business math and linear functions.

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