Problem 13 - Calculus

Question

Find the period and amplitude. $$r=0.1 \sin (\pi t)+2$$

Step-by-Step Solution

Answer

Amplitude: 0.1, Period: 2

Step 1: Identify the General Form

The given equation is \( r = 0.1 \sin(\pi t) + 2 \). The general form of a sinusoidal function is \( r = A \sin(Bt) + C \). Here, \( A = 0.1 \), \( B = \pi \), and \( C = 2 \).

Step 2: Determine the Amplitude

The amplitude of a sinusoidal function \( r = A \sin(Bt) + C \) is given by the absolute value of \( A \). In this case, \( A = 0.1 \), so the amplitude is \( |0.1| = 0.1 \).

Step 3: Calculate the Period

The period of a sine function is calculated using the formula \( \text{Period} = \frac{2\pi}{B} \). Here, \( B = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).

Key Concepts

Sinusoidal Functions

A sinusoidal function is a mathematical function that describes a smooth wave-like pattern. These functions can represent natural phenomena like sound waves, light waves, and tides. They are usually expressed in the form of sine or cosine functions. In the equation provided in the exercise, the sinusoidal function is represented as:

• \( r = A \sin(Bt) + C \)

The variable \( t \) represents time or another independent variable, and \( r \) represents the dependent variable being studied.
The general components of a sinusoidal function include:

• Amplitude (\( A \)): This affects the height of the wave.
• Frequency (related to \( B \)): This determines how frequently the wave oscillates.
• Phase shift (not present in this exercise): This affects the horizontal shift of the wave.
• Vertical shift (\( C \)): This shifts the wave up or down.

Understanding these components allows us to analyze and extract meaningful information from the function efficiently.

Amplitude

The amplitude of a sinusoidal function is a measure of its height from the center line to its peak. In simpler terms, it tells us how high and low the wave reaches from a middle point.
In mathematics, the amplitude is the absolute value of the coefficient of the sine or cosine function. For the function given in the exercise, \( r = 0.1 \sin(\pi t) + 2 \), the amplitude is determined by the value of \( A \), which is 0.1.

• Mathematically,
  the amplitude is calculated as \( |A| \).

In this instance, the amplitude is \( |0.1| = 0.1 \).This means that the wave will oscillate 0.1 units above and below the midline, represented by the vertical shift \( C = 2 \). Knowing the amplitude is essential, especially in real-world contexts where it might signify the maximum strength or magnitude of an oscillating phenomenon.

Period of a Function

The period of a function is the duration it takes for the function to complete one full cycle of its repeating pattern. For sinusoidal functions like sine and cosine, the period is related to how fast the function oscillates.
To find the period of the sine function \( r = A \sin(Bt) + C \), you use the formula:
\[ \text{Period} = \frac{2\pi}{B} \] where \( B \) is the coefficient of \( t \).

• In the given function \( r = 0.1 \sin(\pi t) + 2 \), \( B = \pi \).
• This calculation results in the function's period being \( \frac{2\pi}{\pi} = 2 \).

This indicates that the wave pattern of this function repeats every 2 units of \( t \). Understanding the period is crucial when analyzing cycles in periodic phenomena, whether these cycles represent time, space, or another dimension. Knowing how to calculate and interpret the period allows one to predict and understand oscillating behaviors more clearly.