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Home » f(2) = 2 + rac12 = rac52. - Silent Sales Machine

f(2) = 2 + rac12 = rac52. - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Slope = $rac{-2}{-1} = 2$. The reflected ray has slope 2 relative to the horizontal.

📌 A micropaleontologist observes that the abundance $A(t)$ of a microfossil species follows $A(t) = 1000 e^{-0.03t} \sin(0.1\pi t)$, where $t$ is time in thousands of years. At what time $t > 0$ does the first local maximum occur?

📌 To find maxima, take derivative: $A'(t) = 1000 [ -0.03 e^{-0.03t} \sin(0.1\pi t) + e^{-0.03t} (0.1\pi) \cos(0.1\pi t) ]$.

📌 Factor: $A'(t) = 1000 e^{-0.03t} [ -0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) ]$.

📌 Set derivative to zero: $-0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) = 0$.

📌 $rac{\sin}{\cos} = rac{0.1\pi}{0.03} \Rightarrow an(0.1\pi t) = rac{10\pi}{3} pprox rac{31.416}{3} pprox 10.472$.

📌 Then $0.1\pi t = rctan(10.472) pprox 1.471$ radians.

📌 So $t pprox rac{1.471}{0.1\pi} pprox rac{1.471}{0.3142} pprox 4.684$.

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