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Home » Simplexity Unlocked: Why 90% of Successful Minds Swear by This Game-Changing Concept - Carbonext

Simplexity Unlocked: Why 90% of Successful Minds Swear by This Game-Changing Concept - Carbonext

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 This function can be expressed in the form $ R\cos\left(\frac{\pi}{12}t - \phi\right) $, where $ R = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 $

📌 So the amplitude is $ \boxed{13} $.

📌 To find the time of maximum:

📌 \tan\phi = \frac{12}{5} \Rightarrow \phi = \arctan\left(\frac{12}{5}\right) \approx 1.176 \text{ radians}

📌 Maximum occurs when the cosine term is 1, i.e.,

📌 \frac{\pi}{12}t - \phi = 0 \Rightarrow t = \frac{12}{\pi} \cdot \phi \approx \frac{12}{\pi} \cdot 1.176 \approx 4.49 \text{ hours}

📌 So the maximum value of $ f(t) $ is $ \boxed{13} $, and it occurs at approximately $ \boxed{4.49} $ hours after midnight, or around 4:29 AM.

📌 Question: A city planner is designing a periodic stormwater drainage system that follows a sinusoidal pattern over the year. The water level is modeled by $ w(t) = 7\sin\left(\frac{\pi}{6}t\right) + 24\cos\left(\frac{\pi}{6}t\right) $, where $ t $ is in months. What is the maximum water level, and when does it first occur?

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