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The amplitude of a sine function is a critical concept in both mathematics and its application to real-world problems. In the context of our exercise, the amplitude represents the maximum variation of the volume of air in an adult's lungs, from the average during a breathing cycle.

Understanding this concept is essential for students as it reflects on how much the function values can increase or decrease from its mean position. Given the generic form of a sine function as \[\[\begin{align*}y = A \sin(B(t - C)) + D\end{align*}\]\],where \[\[\begin{align*}A\end{align*}\]\] is the amplitude, it determines the peak value that the sine wave reaches above or below its central value, which in our case is \[\[\begin{align*}55\end{align*}\]\] cubic inches. The given function has an amplitude of \[\[\begin{align*}24.5\end{align*}\]\], which implies that the volume fluctuates by up to 24.5 cubic inches above and below the average volume during a breathing cycle. Essentially, mastering the concept of amplitude in a sine function enables the student to determine how 'big' the waves of the function are.

The period of a sine function is another fundamental concept when exploring sinusoidal functions. It refers to the time it takes for the function to complete one full cycle of its pattern before repeating itself. Our exercise involves the function \[\[\begin{align*}V(t) = 55 + 24.5 \sin\left(\frac{\pi t}{2} - \frac{\pi}{2}\right)\end{align*}\]\],and determining the period helps us to understand the breathing pattern over time.

For the function \[\[\begin{align*}y = A \sin(B(t - C)) + D\end{align*}\]\],the coefficient \[\[\begin{align*}B\end{align*}\]\] affects the period, specifically, the period \[\[\begin{align*}\text{(Period)} = \frac{2\pi}{B}\end{align*}\]\]. In our example, with \[\[\begin{align*}B = \frac{\pi}{2}\end{align*}\]\], the period is calculated to be \[\[\begin{align*}4\end{align*}\]\] seconds. This is significant for our analysis: it informs us that the individual completes a full breathing cycle every 4 seconds. Consequently, by fully grasping the concept of period in a sine function, students can predict the intervals of repetitive patterns in different contexts, a skill very much applicable in disciplines such as physics, engineering, and even biology, as this exercise demonstrates.

Calculating breaths per minute is a practical application of understanding sinusoidal functions, particularly when assessing respiratory rates in medical or biological studies. In our exercise, this calculation stems from the established period of the sine function describing the air volume in the lungs.

To convert the period into breaths per minute, which is a common metric used in health care settings, we first express the 4-second period in terms of minutes, and then we find the reciprocal. This mathematical translation from periods to frequency (breaths per cycle) is crucial as it moves from theoretical understanding into real-world application.

With a 4-second period, we find that this corresponds to \[\[\begin{align*}\frac{1}{15}\end{align*}\]\] minutes per breath. Taking the reciprocal, we get \[\[\begin{align*}15\end{align*}\]\] breaths per minute. It is this process of translation that facilitates the student's ability to take knowledge from the page to practical use – in this case, quantifying how many times a person breathes in one minute, an important vital sign. Such calculations are not only key to precalculus students but also vital for those in healthcare professions.