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Home » \( 0.70^3 = 0.343 \) - RTA

\( 0.70^3 = 0.343 \) - RTA

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Question: A herpetologist models the population of a rare frog species in Madagascar with the function $ F(t) = \frac{100t}{t^2 + 4} $, where $ t $ is time in years. What is the maximum population growth rate?

📌 Solution: To find the maximum of $ F(t) = \frac{100t}{t^2 + 4} $, we compute its derivative using the quotient rule:

📌 F'(t) = \frac{(100)(t^2 + 4) - 100t(2t)}{(t^2 + 4)^2} = \frac{100(t^2 + 4 - 2t^2)}{(t^2 + 4)^2} = \frac{100(4 - t^2)}{(t^2 + 4)^2}.

📌 Set $ F'(t) = 0 $:

📌 - t^2 = 0 \Rightarrow t^2 = 4 \Rightarrow t = \pm 2.

📌 Since $ t $ represents time (non-negative), we consider only $ t = 2 $. To confirm it's a maximum, observe the sign of $ F'(t) $:

📌 For $ t < 2 $, $ 4 - t^2 > 0 \Rightarrow F'(t) > 0 $,

📌 Thus, $ F(t) $ increases up to $ t = 2 $, then decreases — a maximum at $ t = 2 $. Compute the maximum population growth rate:

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