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?

["Understanding the Sinusoidal Stormwater Drainage Model: Maximum Water Level and Timing", "City planners face the critical challenge of designing effective stormwater drainage systems capable of managing fluctuating water levels throughout the year. When the seasonal water level is modeled by a sinusoidal function, understanding its maximum value and the timing of its first occurrence becomes essential for sustainable urban infrastructure. This article explores the mathematical model $ w(t) = 7\sin\left(\frac{\pi}{6}t\right) + 24\cos\left(\frac{\pi}{6}t\right) $, where $ t $ represents time in months, to determine the maximum water level and when it first reaches this peak.", "### Model Breakdown: From Linear Combination to Amplitude-Phase Form", "The stormwater level is expressed as a linear combination of sine and cosine functions with the same angular frequency:", "$$\nw(t) = 7\sin\left(\frac{\pi}{6}t\right) + 24\cos\left(\frac{\pi}{6}t\right)\n$$", "Such expressions can be rewritten in the amplitude-phase form:\n$$\nw(t) = R\sin\left(\frac{\pi}{6}t + \phi\right)\n$$\nwhere $ R $ is the peak amplitude (maximum water level), and $ \phi $ is the phase shift.", "To compute $ R $, use the identity:\n$$\nR = \sqrt{a^2 + b^2}\n$$\nfor $ w(t) = a\sin(\omega t) + b\cos(\omega t) $. Here, $ a = 7 $, $ b = 24 $, so:", "$$\nR = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25\n$$", "Thus, the maximum water level is $ 25 $ units above baseline.", "### When Does the Maximum First Occur?", "The maximum value of a sine function occurs when its argument equals $ \frac{\pi}{2} $. Therefore, set:", "$$\n\frac{\pi}{6}t + \phi = \frac{\pi}{2}\n$$", "To find $ \phi $, use:", "$$\n\ an\phi = \frac{b}{a} = \frac{24}{7}\n\Rightarrow \phi = \ an^{-1}\left(\frac{24}{7}\right)\n$$", "We solve for $ t $:", "$$\n\frac{\pi}{6}t = \frac{\pi}{2} - \phi\n\Rightarrow t = \frac{6}{\pi} \left( \frac{\pi}{2} - \ an^{-1}\left(\frac{24}{7}\right) \right)\n= 3 - \frac{6}{\pi} \ an^{-1}\left(\frac{24}{7}\right)\n$$", "Now approximate $ \ an^{-1}(24/7) $. Since $ 24/7 \approx 3.4286 $, and $ \ an^{-1}(3.4286) \approx 1.287 $ radians (using calculator), we compute:", "$$\nt \approx 3 - \frac{6}{\pi} \cdot 1.287 \approx 3 - \frac{7.722}{3.1416} \approx 3 - 2.457 \approx 0.543 \ ext{ months}\n$$", "Converting 0.543 months into days (assuming ~30 days per month):\n$$\n0.543 \ imes 30 \approx 16.3 \ ext{ days}\n$$", "Thus, the first occurrence of maximum water level is approximately 16 days into the year, around mid-February.", "### Practical Implications for Urban Design", "The model reveals that water levels oscillate with a period of:", "$$\nT = \frac{2\pi}{\omega} = \frac{2\pi}{\pi/6} = 12 \ ext{ months}\n$$", "confirming an annual cycle—consistent with seasonal rainfall and snowmelt patterns. The peak occurs precisely 6 months after the minimum phase shift due to symmetry of the sine wave, peaking at $ t \approx 0.543 $ months. This precise timing enables planners to reinforce drainage systems or implement temporary flood controls in early February.", "### Conclusion", "Through harmonic analysis, we determined that the sinusoidal stormwater model $ w(t) = 7\sin\left(\frac{\pi}{6}t\right) + 24\cos\left(\frac{\pi}{6}t\right) $ reaches a maximum water level of 25 units, with the first peak occurring approximately 16 days into the year. This insight supports proactive, data-driven design of resilient urban stormwater systems.", "---", "Keywords: stormwater drainage, sinusoidal wave model, water level, maximum water level, peak water level timing, urban planning, mathematical modeling, periodic function, sine cosine combination, $ w(t) = 7\sin\left(\frac{\pi}{6}t\right) + 24\cos\left(\frac{\pi}{6}t\right) $", "Meta Description:\nDiscover the maximum water level (25 units) and first occurrence time in a sinusoidal stormwater model $ w(t) = 7\sin\left(\frac{\pi}{6}t\right) + 24\cos\left(\frac{\pi}{6}t\right) $ using harmonic analysis. Essential for urban drainage design."]