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?

["A physiological researcher studying heart rate variability models its daily pattern with the function:\n$$\nf(t) = 5\cos\left(\frac{\pi}{12}t\right) + 12\sin\left(\frac{\pi}{12}t\right)\n$$\nTo determine the amplitude and the time at which the maximum occurs, we express the function in its equivalent sinusoidal form:\n$$\nf(t) = R\cos\left(\frac{\pi}{12}t - \phi\right)\n$$\nwhere $ R $ is the amplitude and $ \phi $ is the phase shift.", "Step 1: Find the amplitude $ R $:\nUsing the identity $ A\cos\ heta + B\sin\ heta = R\cos(\ heta - \phi) $, where $ R = \sqrt{A^2 + B^2} $, we compute:\n$$\nR = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\n$$\nThus, the amplitude of the function is $ \boxed{13} $.", "Step 2: Determine the time of maximum:\nThe maximum of $ f(t) $ occurs when the cosine term reaches its maximum value of 1, i.e., when:\n$$\n\frac{\pi}{12}t - \phi = 0 \quad \Rightarrow \quad t = \frac{12}{\pi} \cdot \phi\n$$", "To find $ \phi $, we use:\n$$\n\ an\phi = \frac{B}{A} = \frac{12}{5} \quad \Rightarrow \quad \phi = \ an^{-1}\left(\frac{12}{5}\right)\n$$", "Thus, the time at which the maximum occurs is:\n$$\nt = \frac{12}{\pi} \cdot \ an^{-1}\left(\frac{12}{5}\right)\n$$", "This value can be evaluated numerically if needed, but the exact expression is:\n$$\nt = \frac{12}{\pi} \cdot \ an^{-1}\left(\frac{12}{5}\right)\n$$", "Final Answer:\nThe amplitude is $ \boxed{13} $, and the maximum occurs at $ \boxed{t = \frac{12}{\pi} \cdot \ an^{-1}\left(\frac{12}{5}\right)} $."]