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Home » Expanding: \( 600 + 60x + 40x + 4x^2 = 1,056 \) - Aurero

Expanding: \( 600 + 60x + 40x + 4x^2 = 1,056 \) - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 The cable is symmetric about $x = 0$ (center), so towers are at $x = \pm40$.

📌 At $x = 40$: $y = rac{1}{200}(40)^2 + 5 = rac{1600}{200} + 5 = 8 + 5 = 13$.

📌 The lowest point is at $x = 0$: $y = 5$.

📌 Vertical distance = $13 - 5 = 8$ meters.

📌 A software engineer implements a 3D rotation using quaternions. The rotation of $90^\circ$ about the $y$-axis is represented by the quaternion $q = \cos( heta/2) + \sin( heta/2)i$, where $ heta = 90^\circ = rac{\pi}{2}$. What is the scalar part of $q$?

📌 $ heta/2 = \pi/4$, so $\cos(\pi/4) = rac{\sqrt{2}}{2}$, $\sin(\pi/4) = rac{\sqrt{2}}{2}$.

📌 The scalar (real) part of $q$ is $\cos(\pi/4) = rac{\sqrt{2}}{2}$.

📌 #### $rac{\sqrt{2}}{2}$

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