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.