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when is the national championship game - Sourci

📅 May 1, 2026 👤 scraface

May 01, 2026

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📌 Question: A zoologist observes that the number of daily interactions between two species in a reserve is modeled by the function $ I(t) = \frac{t^2 + 4}{t + 2} $, where $ t $ is time in days since the start of the study. Find the slant asymptote of $ I(t) $ as $ t \to \infty $.

📌 Solution: To find the slant asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $, we perform polynomial long division of $ t^2 + 4 $ by $ t + 2 $:

📌 Divide $ t^2 $ by $ t $ to get $ t $. Multiply $ t + 2 $ by $ t $ to get $ t^2 + 2t $. Subtract:

📌 Next, divide $ -2t $ by $ t $ to get $ -2 $. Multiply $ t + 2 $ by $ -2 $ to get $ -2t - 4 $. Subtract:

📌 I(t) = t - 2 + \frac{8}{t + 2}.

📌 As $ t \to \infty $, the term $ \frac{8}{t + 2} \to 0 $, so the slant asymptote is:

📌 \boxed{y = t - 2}

📌 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?

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