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Home » f(x) = -\frac76x^3 + \frac212x^2 - \frac823x + 21 - Abu Waleed Tea

f(x) = -\frac76x^3 + \frac212x^2 - \frac823x + 21 - Abu Waleed Tea

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 Question: A forensic anthropologist uses 3D scanning data to model the trajectory of a broken bone fragment as a vector path $\mathbf{r}(t) = \langle t^2, \ln(t), e^{-t} \rangle$ for $t > 0$. At $t = 1$, find the magnitude of the velocity vector $\|\mathbf{r}'(1)\|$.

📌 A: $\sqrt{2 + 0 + e^{-2}} = \sqrt{2 + e^{-2}}$

📌 B: $\sqrt{2^2 + 1 + e^{-1}} = \sqrt{4 + 1 + \frac{1}{e}}$

📌 C: $\sqrt{4 + 0 + 1} = \sqrt{5}$

📌 D: $\sqrt{1 + 1 + 1} = \sqrt{3}$

📌 Question: A civil engineer models the oscillation of a suspension bridge under wind load with the complex equation $(z - 3i)^8 = -16i$. Find the maximum imaginary part among all roots $z$, and express it in the form $\sin \theta$ for some $\theta \in (0, \pi)$.

📌 A: $\sin\left(\frac{3\pi}{8}\right)$

📌 B: $\sin\left(\frac{\pi}{4}\right)$

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