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📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 A herpetologist is modeling the population growth of a species of endangered reptiles using a quadratic function \( P(t) = at^2 + bt + c \), where \( t \) is time in years. If \( P(t) \) reaches its maximum at \( t = 5 \) years and \( a < 0 \), determine the relationship between \( a, b, \) and \( c \).

📌 The vertex form of a quadratic function \( P(t) = at^2 + bt + c \) gives the time at which the maximum value occurs as \( t = -rac{b}{2a} \). Given that the maximum occurs at \( t = 5 \), we have:

📌 = -rac{b}{2a}

📌 Solving for \( b \), we find:

📌 Since \( a < 0 \), this ensures the parabola opens downwards, confirming a maximum. The value of \( c \) does not affect the location of the vertex or the sign of \( a \), so \( c \) can be any real number. Therefore, the relationship between \( a, b, \) and \( c \) is:

📌 oxed{b = -10a ext{ where } a < 0}

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