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.

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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.