1. **Problem Statement:** A theatre group models revenue with $R = -50(t - 2)(t - 3)$ where $t$ is ticket price. They know revenue is maximized at $t=3$ with $R=450$ and $R=400$ at $t=2$. We need to explain their mistake and find the correct revenue equation. 2. **Analyzing the given revenue equation:** The given equation is $R = -50(t - 2)(t - 3)$. Let's expand it: $$R = -50(t^2 - 5t + 6) = -50t^2 + 250t - 300$$ 3. **Check revenue at $t=3$:** $$R(3) = -50(3)^2 + 250(3) - 300 = -450 + 750 - 300 = 0$$ This contradicts the given revenue of 450 at $t=3$. So the model is incorrect. 4. **Check revenue at $t=2$:** $$R(2) = -50(2)^2 + 250(2) - 300 = -200 + 500 - 300 = 0$$ This contradicts the given revenue of 400 at $t=2$. 5. **Correct approach:** Since revenue is maximized at $t=3$ with $R=450$, the vertex of the parabola is at $(3,450)$. The revenue at $t=2$ is 400. 6. **General quadratic form:** $$R = a(t - h)^2 + k$$ where vertex is $(h,k) = (3,450)$. 7. **Substitute vertex:** $$R = a(t - 3)^2 + 450$$ 8. **Use point $(2,400)$ to find $a$:** $$400 = a(2 - 3)^2 + 450$$ $$400 = a( -1)^2 + 450$$ $$400 = a + 450$$ $$a = 400 - 450 = -50$$ 9. **Correct revenue equation:** $$R = -50(t - 3)^2 + 450$$ 10. **Part b) Profit equation:** Given cost equation: $$C = 600 - 50t$$ Profit $P$ is revenue minus cost: $$P = R - C = [-50(t - 3)^2 + 450] - [600 - 50t]$$ 11. **Simplify profit equation:** $$P = -50(t - 3)^2 + 450 - 600 + 50t = -50(t - 3)^2 + 50t - 150$$ 12. **Find break-even ticket price:** Break-even means profit $P=0$: $$0 = -50(t - 3)^2 + 50t - 150$$ Divide both sides by 50: $$0 = - (t - 3)^2 + t - 3$$ Rewrite: $$ (t - 3)^2 = t - 3$$ 13. **Expand and solve:** $$t^2 - 6t + 9 = t - 3$$ Bring all terms to one side: $$t^2 - 6t + 9 - t + 3 = 0$$ $$t^2 - 7t + 12 = 0$$ 14. **Factor quadratic:** $$t^2 - 7t + 12 = (t - 3)(t - 4) = 0$$ 15. **Solutions:** $$t = 3 \text{ or } t = 4$$ 16. **Interpretation:** The production breaks even at ticket prices $3$ and $4$. Since $3$ is the revenue-maximizing price, $4$ is the other break-even point. **Final answers:** - Correct revenue equation: $$R = -50(t - 3)^2 + 450$$ - Profit equation: $$P = -50(t - 3)^2 + 50t - 150$$ - Break-even ticket prices: $$t = 3 \text{ or } t = 4$$