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 \( c \).
["Title: Modeling Endangered Reptile Populations: How a Herpetologist Uses Quadratic Functions with ( a < 0 )", "When studying endangered reptile species, understanding population dynamics is crucial for effective conservation. A herpetologist modeling the population growth of a rare reptile species uses a quadratic function of the form:", "[\nP(t) = at^2 + bt + c\n]", "where ( t ) represents time in years, and ( a, b, c ) are real constants with the condition ( a < 0 ). This negative coefficient for the squared term reflects a realistic scenario: populations may grow and then decline due to habitat loss, predation, or disease, resulting in a bell-shaped curve rather than unchecked exponential increase.", "A key observation is that the population reaches its peak at ( t = 5 ) years. Since the quadratic function is symmetric and its vertex gives the maximum point when ( a < 0 ), the vertex occurs at:", "[\nt = -\frac{b}{2a}\n]", "Setting this equal to 5 yields:", "[\n-\frac{b}{2a} = 5\n]", "Multiplying both sides by ( 2a ) (note that ( a <br/>\ne 0 ), and since ( a < 0 ), multiplying reverses the inequality direction but here we solve for equality):", "[\n-b = 10a \quad \Rightarrow \quad b = -10a\n]", "This equation establishes the direct relationship between the linear coefficient ( b ) and the quadratic coefficient ( a ): ( b ) is exactly ten times ( a ), and since ( a < 0 ), ( b ) is positive, meaning the population rises sharply at first before declining.", "While ( c ) represents the initial population at ( t = 0 ) (( P(0) = c )), it does not affect the location of the maximum—only the timing and curvature. Therefore, ( c ) is independent of the condition ( a < 0 ) and vertex at ( t = 5 ), though it anchors the model in observed data.", "In conclusion, for a concave-down parabola modeling endangered reptile populations peaking at ( t = 5 ), the essential relationship is:", "[\nb = -10a\n]", "with ( a < 0 ). This quadratic modeling helps conservationists anticipate critical population thresholds and plan timely interventions to prevent extinction.", "---", "Keywords: herpetologist, population modeling, quadratic function, endangered reptiles, ( P(t) = at^2 + bt + c ), vertex at ( t = 5 ), ( a < 0 ), conservation biology."]
