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Home » A function \(f(x) = racx^2 - 4x - 2\) is defined for \(x - inBeat

A function \(f(x) = racx^2 - 4x - 2\) is defined for \(x - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 To find the time at which this maximum occurs, we use:

📌 \tan\phi = \frac{3}{4} \Rightarrow \phi = \arctan\left(\frac{3}{4}\right)

📌 The maximum of $ \sin(\cdot) $ occurs when its argument is $ \frac{\pi}{2} $, so:

📌 \frac{\pi}{6}t + \phi = \frac{\pi}{2} \Rightarrow t = \frac{\frac{\pi}{2} - \phi}{\frac{\pi}{6}} = \frac{6}{\pi} \left( \frac{\pi}{2} - \arctan\left(\frac{3}{4}\right) \right)

📌 Using $ \arctan\left(\frac{3}{4}\right) \approx 0.6435 $ radians:

📌 t \approx \frac{6}{\pi} \left( \frac{\pi}{2} - 0.6435 \right) \approx \frac{6}{\pi} (1.5708 - 0.6435) = \frac{6}{\pi} (0.9273) \approx 1.77 \text{ hours}

📌 So the maximum tide height of $ \boxed{5} $ occurs approximately at $ \boxed{1.77} $ hours after midnight, or around 1:46 AM.

📌 Question: A physiological researcher is analyzing the periodic response of heart rate variability over a 24-hour period, modeled by the function $ f(t) = 5\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right) $. What is the amplitude of this function, and at what time does the maximum occur?

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