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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Question: An epidemiologist models the spread of a virus in a population with the equation $ p(t) = -t^2 + 14t + 30 $, where $ p(t) $ represents the number of infected individuals at time $ t $. What is the maximum number of infected individuals?

📌 Solution: The quadratic $ p(t) = -t^2 + 14t + 30 $ opens downward. The vertex occurs at $ t = -\frac{b}{2a} = -\frac{14}{2(-1)} = 7 $. Substituting $ t = 7 $, $ p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79 $.

📌 \boxed{79}

📌 Question: A museum curator arranges two exhibit pathways modeled by the lines $ y = 4x - 5 $ and $ 2y + 3x = 10 $. Find their intersection point.

📌 Solution: Substitute $ y = 4x - 5 $ into the second equation: $ 2(4x - 5) + 3x = 10 $. Simplify: $ 8x - 10 + 3x = 10 \Rightarrow 11x = 20 \Rightarrow x = \frac{20}{11} $. Substitute back: $ y = 4\left(\frac{20}{11}\right) - 5 = \frac{80}{11} - \frac{55}{11} = \frac{25}{11} $.

📌 \boxed{\left( \frac{20}{11}, \frac{25}{11} \right)}

📌 Question: Expand the product $ (5a - 3b)(2a + 7b) $.

📌 Solution: Use the distributive property:

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