Problem 49 - Algebra 2

Question

The carrier wave for a certain FM radio station can be modeled by the equation \(y=A \sin \left(10^{7} \cdot 2 \pi t\right),\) where \(A\) is the amplitude of the wave and \(t\) is the time in seconds. Determine the period of the carrier wave.

Step-by-Step Solution

Answer

The period of the carrier wave is \( 10^{-7} \) seconds.

Step 1: Understanding the Problem

We need to determine the period of the given sine function equation, \( y = A \sin (10^7 \cdot 2 \pi t) \). The general form of a sine wave is \( y = A \sin (2\pi f t) \), where \( f \) is the frequency. The coefficient of \( t \) in the sine function helps us find the frequency of the wave.

Step 2: Identifying the Frequency

In the given equation, the coefficient of \( t \) inside the sine function is \( 10^7 \cdot 2\pi \). This represents the angular frequency \( \omega = 2\pi f \). Hence, we can equate \( 2\pi f = 10^7 \cdot 2\pi \) to find the frequency \( f \).

Step 3: Solving for Frequency

To find the frequency \( f \), divide both sides of the equation \( 2\pi f = 10^7 \cdot 2\pi \) by \( 2\pi \), resulting in \( f = 10^7 \). Therefore, the frequency of the wave is \( 10^7 \) Hz.

Step 4: Calculate the Period

The period \( T \) of a wave is the reciprocal of its frequency \( f \). Thus, using \( T = \frac{1}{f} \), we find the period \( T = \frac{1}{10^7} = 10^{-7} \) seconds.

Key Concepts

Period of a Wave

The period of a wave is a fundamental concept in wave mechanics. It represents the time it takes for one complete cycle of a wave to pass a given point. For any wave modeled by a function like a sine wave, the period is inversely related to the wave's frequency.
For a wave expressed as \( y = A \sin(2\pi f t) \), you can determine the period, denoted as \( T \), using the formula \( T = \frac{1}{f} \). Here, \( f \) stands for the frequency of the wave.
The periodic nature of waves is essential in various applications including radio, where exact periods allow for the tuning of specific frequencies. Calculating the period helps understand signal behavior and the timing of oscillations.

Sine Function

The sine function is a crucial tool in modeling waves, particularly in physics and engineering. It is widely used because it naturally describes oscillating phenomena. The general form of a sine wave is \( y = A \sin(2\pi f t) \).
When using the sine function:

• **\( A \)** is the amplitude, indicating the maximum height of the wave from its equilibrium (zero) position.
• **\( 2\pi f \)** represents the angular frequency, dictating how rapidly the function oscillates over time \( t \).

This mathematical function repeats its values in regular intervals, making it perfect for representing periodic waves like those found in sound and radio signals. By adjusting parameters, sine waves can illustrate various real-world behaviors.

Frequency of a Wave

Frequency is a core concept in wave dynamics, referring to how many complete cycles of the wave occur per second. In mathematical terms, it is measured in hertz (Hz), with one hertz equal to one cycle per second.
In an expression \( y = A \sin(2\pi f t) \), the term \( f \) stands for the frequency. Increasing the frequency of a wave means the waves occur more frequently, resulting in shorter periods. This is reflected in the relationship \( T = \frac{1}{f} \), where \( T \) is the period.

• Frequency and period are inversely related: as frequency goes up, the period decreases, and vice versa.
• High frequency waves are often used in communications, including FM radio, to allow for clear signal transmission.

Understanding frequency is vital for analyzing waves and predicting how they behave over time.