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Home » 8192 → 4096 (2) - RoadRUNNER Motorcycle Touring & Travel Magazine

8192 → 4096 (2) - RoadRUNNER Motorcycle Touring & Travel Magazine

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 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?

📌 We combine the expression into a single sine function:

📌 w(t) = R\sin\left(\frac{\pi}{6}t + \phi\right)

📌 where $ R = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 $

📌 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:

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