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In the uploaded image, you are dealing with a production problem involving a cost function \( C(x) \) and a revenue function \( R(x) \). Here's the given information:

- **Cost Function**: \( C(x) = 270x + 9630 \), where \( x \) is the number of units produced.
- **Revenue Function**: \( R(x) = 360x \).

You are asked to:
1. Find and analyze the break-even quantity.
2. Write the profit function \( P(x) \).

### Step 1: Break-even Quantity
The break-even point occurs when the cost and revenue are equal, meaning \( C(x) = R(x) \).

Let's solve for \( x \):

\[
270x + 9630 = 360x
\]

Subtract \( 270x \) from both sides:

\[
9630 = 360x - 270x
\]

\[
9630 = 90x
\]

Now, divide both sides by 90:

\[
x = \frac{9630}{90} = 107
\]

Thus, the break-even quantity is **107 units**.

### Step 2: Profit Function
The profit function is defined as the difference between the revenue and the cost:

\[
P(x) = R(x) - C(x)
\]

Substitute the given functions for \( R(x) \) and \( C(x) \):

\[
P(x) = (360x) - (270x + 9630)
\]

Simplify:

\[
P(x) = 360x - 270x - 9630
\]

\[
P(x) = 90x - 9630
\]

So, the profit function is:

\[
P(x) = 90x - 9630
\]

### Conclusion
1. The break-even quantity is **107 units**.
2. The profit function is \( P(x) = 90x - 9630 \).

Would you like more details or explanations?

Here are 5 related questions you might find useful:
1. What happens if the production level is exactly at the break-even point?
2. How does the profit change when producing 110 units instead of 107?
3. What is the revenue when producing 45 units?
4. Can the company make a profit if they produce 50 units?
5. How would changes in the cost per unit affect the break-even quantity?

**Tip:** Always check the intersection between cost and revenue functions for analyzing profit margins.

A firm produces a product that has the production cost function C(x) = 270x + 9630 and the revenue function R(x) = 360x. No more than 45 units can be sold. Find and analyze the break-even quantity, then find the profit function.

Analyze a firm's break-even point with a cost function C(x) = 270x + 9630 and revenue function R(x) = 360x. Calculate the break-even quantity and the profit function P(x) = 90x - 9630. Understand the key concepts of break-even analysis and profit calculation for high school students (Grades 10-12).

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