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📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 A chemist is analyzing a reaction involving a catalyst and models the concentration of a reactant over time \(t\) with the function \(C(t) = \frac{kt}{t^2 + 1}\). Determine the time \(t\) at which the concentration is maximized.

📌 To find when \(C(t) = \frac{kt}{t^2 + 1}\) is maximized, find the critical points by setting the derivative \(C'(t)\) to zero.

📌 First, compute the derivative using the quotient rule:

📌 \[ C(t) = \frac{kt}{t^2 + 1} \]

📌 If \(u = kt\) and \(v = t^2 + 1\), then

📌 \[ u' = k, \quad v' = 2t \]

📌 \[ C'(t) = \frac{(k)(t^2 + 1) - (kt)(2t)}{(t^2 + 1)^2} \]

📌 \[ C'(t) = \frac{k(t^2 + 1) - 2kt^2}{(t^2 + 1)^2} \]

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