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Home » \fracdVdt = 4\pi r^2 \fracdrdt - Treasure Valley Movers

\fracdVdt = 4\pi r^2 \fracdrdt - Treasure Valley Movers

📅 April 17, 2026 👤 scraface

Apr 17, 2026

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📌 Question: A game theory economist analyzes the payoff function $ P(x) = -x^2 + 4x + m $. What value of $ m $ ensures the maximum payoff is 12?

📌 Solution: The maximum of $ P(x) = -x^2 + 4x + m $ occurs at the vertex. The $ x $-coordinate of the vertex is $ x = \frac{-b}{2a} = \frac{-4}{-2} = 2 $. Substitute $ x = 2 $ into $ P(x) $:

📌 = -(2)^2 + 4(2) + m

📌 = -4 + 8 + m

📌 The value of $ m $ is $ \boxed{8} $.

📌 Question: A science communicator models the growth of a YouTube channel's subscribers with $ S(t) = at^2 + bt + c $. If $ S(1) = 500 $, $ S(2) = 1200 $, and $ S(3) = 2100 $, find $ a $.

📌 Solution: Set up equations using the given points:

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