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Home » I’m Heading Out Tonight—You Won’t BELIEVE What I Found Outside! - ToelettAPP

I’m Heading Out Tonight—You Won’t BELIEVE What I Found Outside! - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 So the maximum water level is $ \boxed{25} $.

📌 To find the first time it occurs:

📌 \tan\phi = \frac{24}{7} \Rightarrow \phi = \arctan\left(\frac{24}{7}\right) \approx 1.287 \text{ radians}

📌 Maximum occurs when:

📌 \frac{\pi}{6}t + \phi = \frac{\pi}{2} \Rightarrow t = \frac{6}{\pi} \left( \frac{\pi}{2} - \phi \right) = \frac{6}{\pi} \left( \frac{\pi}{2} - 1.287 \right)

📌 t \approx \frac{6}{\pi} (1.571 - 1.287) = \frac{6}{\pi} (0.284) \approx 0.54 \text{ months}

📌 So the maximum water level is $ \boxed{25} $, and it first occurs approximately $ \boxed{0.54} $ months after the start of the year, or around early January.

📌 Question: A disaster response team models the periodic power grid stability using the function $ p(t) = 9\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right) $. What is the amplitude and the phase shift $ \phi $ in the equivalent form $ R\cos\left(\frac{\pi}{12}t - \phi\right) $?

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