A firm produces a product that has the production cost function 

​C(x)equals=340x+9520 

and the revenue function 

​R(x)=425x. 

No more than 

261

units can be sold. Find and analyze the​ break-even quantity, then find the profit function. 

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.

Learn how to calculate break-even quantity and profit functions using cost and revenue functions. Understand the relationship between production costs and revenue generation. Example problem: Firm produces a product with cost function C(x) = 340x + 9520 and revenue function R(x) = 425x.

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