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Home » You Won’t Believe How 4 and Inches Changes Your Life Forever! - Simpleprint

You Won’t Believe How 4 and Inches Changes Your Life Forever! - Simpleprint

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: A pharmacologist is modeling the concentration of a drug in the bloodstream over time with the function $ C(t) = \frac{3t}{t^2 + 4} $. Find the time $ t \geq 0 $ at which the concentration is maximized.

📌 Solution: To maximize $ C(t) = \frac{3t}{t^2 + 4} $, we take its derivative using the quotient rule:

📌 C'(t) = \frac{(3)(t^2 + 4) - 3t(2t)}{(t^2 + 4)^2} = \frac{3t^2 + 12 - 6t^2}{(t^2 + 4)^2} = \frac{-3t^2 + 12}{(t^2 + 4)^2}.

📌 Set $ C'(t) = 0 $:

📌 -3t^2 + 12 = 0 \Rightarrow t^2 = 4 \Rightarrow t = 2 \quad (\text{since } t \geq 0).

📌 To confirm this is a maximum, note that $ C'(t) > 0 $ for $ 0 < t < 2 $ and $ C'(t) < 0 $ for $ t > 2 $, so the function increases then decreases. Thus, the maximum occurs at $ t = 2 $.

📌 Question: A linguist is analyzing sentence structures and models the complexity of a sentence as $ f(n) = \frac{n^2 + 2n + 3}{n + 1} $, where $ n $ is the number of clauses. Find the minimum value of $ f(n) $ for $ n > 0 $.

📌 Solution: Simplify the function:

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