Let's go through the steps to find and analyze the break-even quantity, as well as the profit function.

Step 1: Understand the Given Functions

• Cost Function: $C(x) = 340x + 9520$
• Revenue Function: $R(x) = 425x$

Here, $x$ represents the number of units produced and sold.

Step 2: Find the Break-Even Quantity

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

Set the revenue function equal to the cost function: $425x = 340x + 9520$

Step 3: Solve for $x$

Subtract $340x$ from both sides to isolate the variable: $425x - 340x = 9520$ $85x = 9520$ $x = \frac{9520}{85} = 112$

So, the break-even quantity is $x = 112$ units.

Step 4: Analyze the Break-Even Quantity

At $x = 112$, the firm neither makes a profit nor a loss. If the firm produces and sells fewer than 112 units, it incurs a loss. If it sells more than 112 units, it makes a profit.

Step 5: Find the Profit Function

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

Substitute the given functions: $P(x) = 425x - (340x + 9520)$

Distribute and simplify: $P(x) = 425x - 340x - 9520$ $P(x) = 85x - 9520$

So, the profit function is: $P(x) = 85x - 9520$

Step 6: Determine the Profit at Maximum Sales

Given that no more than 261 units can be sold, we can evaluate the profit at this maximum quantity:

$P(261) = 85(261) - 9520$ $P(261) = 22185 - 9520 = 12665$

Thus, if the firm sells the maximum of 261 units, the profit will be $12,665.

Summary

• Break-even quantity: $x = 112$ units.
• Profit function: $P(x) = 85x - 9520$.
• Maximum possible profit (at 261 units): $12,665.

Would you like more details or have any questions?

Related Questions:

1. What happens to the profit if the cost per unit increases by $10?
2. How would a change in fixed costs affect the break-even quantity?
3. What is the revenue at the break-even point?
4. How does the break-even analysis change if the price per unit is reduced?
5. How would changes in demand (e.g., selling fewer than 261 units) impact profit?

Tip:

Understanding the break-even point helps businesses decide on production levels. Beyond this point, each additional unit sold contributes to profit.