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Home » Thus, the area of the path is \(\boxed44\pi\) square meters. - DNSFLEX

Thus, the area of the path is \(\boxed44\pi\) square meters. - DNSFLEX

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 4Question: A chemist models the efficiency of a new green catalyst with the function $ f(x) = x^2 - 6x + k $, where $ x $ represents reaction temperature and $ k $ is a parameter related to material properties. If the catalyst achieves peak efficiency when $ x = 3 $, what is the value of $ k $ such that $ f(3) = 0 $?

📌 Solution: To determine the value of $ k $, substitute $ x = 3 $ into the function and set $ f(3) = 0 $:

📌 f(3) = (3)^2 - 6(3) + k = 0

📌 - 18 + k = 0

📌 Thus, the value of $ k $ is $\boxed{9}$.

📌 Question: A zoologist models the population growth of a rare species in the Amazon with the function $ P(t) = t^2 + bt + 12 $, where $ t $ is time in years. If the population stabilizes (i.e., reaches zero growth) at $ t = -6 $, what is the value of $ b $?

📌 Solution: Since the population stabilizes at $ t = -6 $, this means $ P(-6) = 0 $. Substituting into the equation:

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