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the palms oceanfront hotel isle of palms - Anadea

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 Solution: A regular octahedron has six vertices, with pairs at $ (\pm1, 0, 0) $, $ (0, \pm1, 0) $, and $ (0, 0, \pm1) $. Given three vertices, the missing vertex must be $ (0, 0, 1) $ or $ (0, 0, -1) $. Since the problem specifies integer coordinates, both are valid, but the fourth vertex (completing the octahedron) is $ (0, 0, 1) $.

📌 \boxed{(0, 0, 1)}

📌 Question: Compute $ \left( \cos 45^\circ + i \sin 45^\circ \right)^8 $ using De Moivre’s Theorem.

📌 Solution: By De Moivre’s Theorem, $ \left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta) $. Here, $ \theta = 45^\circ $, $ n = 8 $, so:

📌 $ \cos(360^\circ) + i \sin(360^\circ) = 1 + i(0) = 1 $.

📌 Question: A UX researcher analyzes user engagement metrics modeled by $ \log_2(x + 3) = 4 $. Find the value of $ x $.

📌 Solution: Rewrite the logarithmic equation in exponential form: $ x + 3 = 2^4 $. Simplify $ 2^4 = 16 $, so $ x = 16 - 3 = 13 $.

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