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Home » We want when ratio was 1:1, i.e., 1 - Londonproperty

We want when ratio was 1:1, i.e., 1 - Londonproperty

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Simplify numerator:

📌 f'(u) = \frac{(1 + 9u)(1 - 9u)}{(1 + 9u)^4} = \frac{1 - 81u^2}{(1 + 9u)^3}.

📌 Set $ f'(u) = 0 \Rightarrow 1 - 81u^2 = 0 \Rightarrow u^2 = \frac{1}{81} \Rightarrow u = \frac{1}{9} $ (since $ u > 0 $).

📌 Recall $ u = e^{-0.5t} $, so:

📌 e^{-0.5t} = \frac{1}{9} \Rightarrow -0.5t = \ln\left(\frac{1}{9}\right) = -\ln 9 \Rightarrow t = 2\ln 9.

📌 Thus, the time at which the rate of change is maximized is:

📌 Question: A deep-sea microbial population grows such that the number of organisms at time $ t $ is modeled by

📌 N(t) = \frac{t^3 + 2t^2 - 5t + 6}{t - 1}.

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