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Home » Volume = (4/3) * 3,14 * 125 = 523,33 cm³ - Red Crowns

Volume = (4/3) * 3,14 * 125 = 523,33 cm³ - Red Crowns

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \theta = 10^\circ, 110^\circ, 130^\circ, 230^\circ, 250^\circ, 350^\circ

📌 The maximum value is $ \boxed{350^\circ} $. This corresponds to the peak resonance angle in the glacial oscillation model when physical constraints limit the domain to this interval.

📌 \boxed{350^\circ}

📌 Question: Compute $ \left( \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} \right)^{14} + \left( \cos \frac{2\pi}{7} - i \sin \frac{2\pi}{7} \right)^{14} $, a quantity arising in modeling periodic synthetic microbial interactions in subglacial biofilm networks.

📌 Let $ z = \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} = e^{i \frac{2\pi}{7}} $. Then its conjugate is $ \overline{z} = \cos \frac{2\pi}{7} - i \sin \frac{2\pi}{7} = e^{-i \frac{2\pi}{7}} $.

📌 z^{14} + \overline{z}^{14} = \left(e^{i \frac{2\pi}{7}}\right)^{14} + \left(e^{-i \frac{2\pi}{7}}\right)^{14} = e^{i 4\pi} + e^{-i 4\pi}

📌 Since $ e^{i 4\pi} = \cos(4\pi) + i \sin(4\pi) = 1 $, and similarly $ e^{-i 4\pi} = 1 $, we get:

📌 z^{14} + \overline{z}^{14} = 1 + 1 = 2

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