1. **Problem Statement:** You asked why the term $50t$ was added when simplifying the profit equation in problem b. 2. **Recall the profit equation:** Profit $P$ is revenue minus cost: $$P = R - C$$ Given: $$R = -50(t - 3)^2 + 450$$ $$C = 600 - 50t$$ 3. **Substitute $R$ and $C$ into $P$:** $$P = [-50(t - 3)^2 + 450] - [600 - 50t]$$ 4. **Distribute the minus sign over the cost:** $$P = -50(t - 3)^2 + 450 - 600 + 50t$$ 5. **Explanation:** When subtracting the entire cost expression, the minus sign applies to both terms inside the brackets: - The $600$ becomes $-600$ - The $-50t$ becomes $+50t$ because $-(-50t) = +50t$ 6. **Simplify constants:** $$450 - 600 = -150$$ 7. **Final simplified profit equation:** $$P = -50(t - 3)^2 + 50t - 150$$ **Summary:** The $50t$ term appears because subtracting $-50t$ is equivalent to adding $50t$. This is a standard rule of algebra when subtracting expressions inside parentheses.