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Home » The radius of the sphere is 5 centimeters. - Dachbleche24

The radius of the sphere is 5 centimeters. - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 A zoologist observes that the population \( P \) of a certain species in the Amazon can be modeled by the equation \( P = \frac{500}{1 + e^{-0.1(t-5)}} \), where \( t \) is the number of months since the study began. Calculate the population of the species as \( t \) approaches infinity.

📌 To determine the population \( P \) as \( t \to \infty \), we analyze the behavior of the logistic function:

📌 \[ P = \frac{500}{1 + e^{-0.1(t-5)}} \]

📌 As \( t \to \infty \), the term \( e^{-0.1(t-5)} \to 0 \). Thus, the denominator approaches 1:

📌 \[ P \to \frac{500}{1 + 0} = 500 \]

📌 Therefore, the population approaches \(\boxed{500}\) as \( t \to \infty \).

📌 A health data analyst models patient recovery rates with the function \( R(x) = \ln(x^2 + 4x + 5) \). Find the value of \( x \) that maximizes \( R(x) \).

📌 To maximize \( R(x) = \ln(x^2 + 4x + 5) \), we first find the derivative \( R'(x) \) and set it to zero.

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