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Home » Total time: \(3 + 4 = 7\) hours - Midis

Total time: \(3 + 4 = 7\) hours - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 so $ t \geq 2 $. The expression becomes $ t + rac{1}{t} $. Define $ f(t) = t + rac{1}{t} $ for $ t \geq 2 $. The derivative $ f'(t) = 1 - rac{1}{t^2} > 0 $ for $ t > 1 $, so $ f(t) $ is increasing. Therefore, the minimum occurs at $ t = 2 $:

📌 f(2) = 2 + rac{1}{2} = rac{5}{2}.

📌 Equality holds when $ x = y $, which satisfies $ x, y > 0 $. Thus, the minimum value is

📌 oxed{rac{5}{2}}.

📌 Question: A hydrologist models the flow resistance $ R $ through a porous aquifer as

📌 R = rac{p + q}{p - q} + rac{p - q}{p + q} - 2,

📌 where $ p > q > 0 $ are hydraulic conductivity parameters. Find the minimum value of $ R $.

📌 Solution: Let $ a = rac{p + q}{p - q} $ and $ b = rac{p - q}{p + q} $. Note that $ b = rac{1}{a} $, and since $ p > q > 0 $, $ a > 1 $. Then

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