https://file.aiquickdraw.com/mathbot/file/1730087450647rkhbbzoy.jpg

Let's solve each problem in turn.

### Problem 11:
The break-even point occurs when cost \( C(x) \) equals revenue \( R(x) \), so we set \( C(x) = R(x) \) and solve for \( x \).

Given:
\[
C(x) = 17x + 24,000
\]
\[
R(x) = 25x
\]

To find the break-even point, set \( C(x) = R(x) \):
\[
17x + 24,000 = 25x
\]
Subtract \( 17x \) from both sides:
\[
24,000 = 8x
\]
Now, divide by 8:
\[
x = 3,000
\]

To find the \( y \)-value, substitute \( x = 3,000 \) into either \( C(x) \) or \( R(x) \). Using \( R(x) = 25x \):
\[
R(3,000) = 25 \times 3,000 = 75,000
\]

So, the break-even point is:
\[
(x, y) = (3,000, 75,000)
\]

### Problem 12:
Similarly, set \( C(x) = R(x) \) and solve for \( x \).

Given:
\[
C(x) = 90x + 16,000
\]
\[
R(x) = 210x
\]

Set \( C(x) = R(x) \):
\[
90x + 16,000 = 210x
\]
Subtract \( 90x \) from both sides:
\[
16,000 = 120x
\]
Now, divide by 120:
\[
x = 133.33
\]

Since we are asked to round to the nearest whole number, \( x = 133 \).

To find the \( y \)-value, substitute \( x = 133 \) into either \( C(x) \) or \( R(x) \). Using \( R(x) = 210x \):
\[
R(133) = 210 \times 133 = 27,930
\]

So, the break-even point is:
\[
(x, y) = (133, 27,930)
\]

### Final Answers:
- Problem 11: \( (x, y) = (3,000, 75,000) \)
- Problem 12: \( (x, y) = (133, 27,930) \)

Would you like more detailed steps or have any questions?

---

1. How is the break-even point derived in business contexts?
2. What are other applications of break-even analysis?
3. How does changing costs or prices impact the break-even point?
4. What is the importance of rounding in business calculations?
5. Why is it sometimes better to use revenue functions over profit functions?

**Tip:** For break-even analysis, always check if costs include fixed and variable components as it affects the calculation.

Find the break-even point for the firm whose cost function C and revenue function R are given. For Problem 11: C(x) = 17x + 24,000, R(x) = 25x. For Problem 12: C(x) = 90x + 16,000, R(x) = 210x. Round your answers to the nearest whole number.

Learn how to find the break-even points for firms using given cost and revenue functions in a step-by-step algebraic approach. This problem is ideal for students in Grades 9-12 who want to understand linear equations and break-even analysis.

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