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.