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Home » Is the Pyramid Head the Key to Unlocking the Lost Civilizations? Discover Now! - Simpleprint

Is the Pyramid Head the Key to Unlocking the Lost Civilizations? Discover Now! - Simpleprint

📅 March 13, 2026 👤 scraface

Mar 13, 2026

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📌 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?

📌 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.

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