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Home » The circumference formula is \(C = 2\pi r\). - go-checkin.com

The circumference formula is \(C = 2\pi r\). - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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 $\cos(\pi/4) = rac{\sqrt{2}}{2}$.

📌 #### $rac{\sqrt{2}}{2}$

📌 A micropaleontologist uses carbon-14 dating on a foraminifera sample. If the sample has 35% of its original carbon-14 remaining, and the half-life of carbon-14 is 5730 years, how old is the sample? Use $N = N_0 e^{-kt}$, $k = rac{\ln 2}{5730}$.

📌 Given $N/N_0 = 0.35 = e^{-kt}$, so $\ln(0.35) = -kt$.

📌 $t = -rac{\ln(0.35)}{k} = -rac{\ln(0.35) \cdot 5730}{\ln(2)}$.

📌 Calculate: $\ln(0.35) pprox -1.0498$, $\ln(2) pprox 0.6931$, so $t pprox rac{1.0498}{0.6931} \cdot 5730 pprox 1.5147 \cdot 5730 pprox 8679$.

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