Problem 71 - Precalculus : Building Concepts and Connections

Question

The function $$P(t)=50 \sin \frac{2 \pi}{23} t+50$$ is used in biorhythm theory to predict an individual's physical potential (as a percentage of the maximum) on a particular day, with \(t=0\) corresponding to birth. (a) What is the period of the function? (b) What is an individual's physical potential on her or his third birthday (day \(1095) ?\)

Step-by-Step Solution

Answer

The period of the function is 23 days. The physical potential on the third birthday can be found by substituting \(t=1095\) into the function and calculating the result.

Step 1: Find the Period of the Function

The period of a sinusoidal function of the form \(P(t)=A \sin (Bt+C)+D\) is found by calculating \(\frac{2\pi}{|B|}\). In this case, the function is \(P(t)=50 \sin \left(\frac{2\pi}{23} t\right)+50\), so \(B=\frac{2\pi}{23}\). Thus, the period of the function is \(\frac{2\pi}{\frac{2\pi}{23}}=23\)

Step 2: Find the Physical Potential on the Third Birthday

To find the physical potential on the third birthday (1095 days), substitute \(t = 1095\) in the equation and calculate \(P(1095)\). In other words, find the value of \(50 \sin \left(\frac{2\pi}{23} \times 1095\right) + 50\) using a calculator.

Step 3: Interpret the Results

The period of the function gives the number of days after which the person's physical potential cycle repeats itself, in this case every 23 days. Moreover, \(P(1095)\) denotes the individual's physical potential on their third birthday as a percentage of their maximum potential.

Key Concepts

Sinusoidal Function

A sinusoidal function is a type of periodic function. It repeats its values at regular intervals or periods. These functions often model natural phenomena, including daily temperature variations and even human biorhythms.

• The basic formula for a sinusoidal function is: \[P(t) = A \sin(Bt + C) + D\]
• Here, \(A\) is the amplitude, indicating the function's range or peak height. It dictates how high and low the wave goes.
• The parameter \(B\) affects the period of the function, which is how often the function repeats itself.
• \(C\) is the phase shift, determining how the wave moves horizontally along the x-axis.
• \(D\) is the vertical shift, which moves the entire function up or down.

In the given biorhythm function, \[P(t)=50 \sin \frac{2 \pi}{23} t+50\],- The amplitude is 50, meaning the physical potential ranges from 0 to 100.- The entire sinusoidal wave is shifted up by 50 units, ensuring physical potential is always positive.This is a great representation of how sinusoidal functions can model various cycles in life.

Period of Function

The period of a function is the length of one complete cycle of a repeating function. Simply put, it's the smallest interval after which the function starts repeating itself with the same pattern.

• For a sinusoidal function defined as \(P(t) = A \sin(Bt + C) + D\), the period \(T\) is calculated by the formula: \[T = \frac{2\pi}{|B|}\]
• In this biorhythm function example, \[B = \frac{2\pi}{23},\] so the period is \[T = \frac{2\pi}{\frac{2\pi}{23}} = 23\] days.

This means every 23 days, the physical potential function completes a full cycle and starts again from the beginning. Understanding the period helps predict patterns and cycles inherent in many natural and human-made systems.

Physical Potential Calculation

To find the physical potential, especially for special occasions like birthdays as in the exercise, involves substituting specific values of \(t\) (like specific days) into the sinusoidal function.For instance, to calculate physical potential on the third birthday:

• Here, \(t = 1095\) days (as there are 365 days in a year multiplied by 3).
• You substitute \(t = 1095\) into the function \[P(t)=50 \sin \left(\frac{2\pi}{23} \times 1095\right) + 50\].
• Use a calculator to determine the value of \[50 \sin \left(\frac{2\pi}{23} \times 1095\right) + 50\].
• This gives the physical potential as a percentage.

The result is the potential in terms of a percentage. It reveals the body's predicted energy level, based on the cyclic nature of these functions. Understanding these calculations can shed light on cyclic fluctuations in energy or capability.