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Home » Thus, valid arrangements: \(120 - 48 = 72\) - Crosslake

Thus, valid arrangements: \(120 - 48 = 72\) - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 A science fiction writer models the growth of a resource-dependent colony on Mars using the equation \( a + b = 12 \) and \( a^2 + b^2 = 80 \). Find \( a^3 + b^3 \).

📌 a + b = 12 \quad \text{and} \quad a^2 + b^2 = 80

📌 Use the identity:

📌 We know \( (a + b)^3 = 12^3 = 1728 \). To use the identity, we need \( ab \). From the identity:

📌 = 12^2 - 2ab = 144 - 2ab

📌 Solve for \( ab \):

📌 2ab = 144 - 80 = 64 \quad \Rightarrow \quad ab = 32

📌 Now substitute into the formula for \( a^3 + b^3 \):

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