Problem 28 - Precalculus Mathematics for Calculus

Question

FM Radio Signals The carrier wave for an FM radio signal is modeled by the function $$ y=a \sin \left(2 \pi\left(9.15 \times 10^{7}\right) t\right) $$ where \(t\) is measured in seconds. Find the period and frequency of the carrier wave.

Step-by-Step Solution

Answer

The frequency is \( 9.15 \times 10^{7} \) Hz, and the period is approximately \( 1.093 \times 10^{-8} \) seconds.

Step 1: Understanding the Given Function

The function given for the FM radio signal is \( y = a \sin(2\pi(9.15 \times 10^{7})t) \). This is in the form of \( y = a \sin(2\pi ft) \), where \( f \) is the frequency of the wave. From this, we can see that the frequency \( f \) is \( 9.15 \times 10^{7} \) Hz.

Step 2: Calculate the Period Using the Frequency

The period \( T \) of a wave is the reciprocal of the frequency. Mathematically, this can be expressed as \( T = \frac{1}{f} \). Substitute \( f = 9.15 \times 10^{7} \) Hz into this equation: \[ T = \frac{1}{9.15 \times 10^{7}} \text{ seconds} \approx 1.093 \times 10^{-8} \text{ seconds} \]

Step 3: Conclude with the Frequency and Period

Now we have both the frequency and the period of the carrier wave. The frequency of the carrier wave is \( 9.15 \times 10^{7} \) Hz (or 91.5 MHz), and the period is approximately \( 1.093 \times 10^{-8} \) seconds.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions in mathematics. It is often used to model periodic phenomena, such as sound waves or alternating current waveforms. The general form of a sine function is given by:\[y = a \, \sin(2 \pi f t + \phi) \]Here:- \(a\) is the amplitude, which determines the height of the waves.- \(f\) is the frequency, indicating how many cycles occur per second.- \(t\) is the time variable.- \(\phi\) is the phase shift, which moves the wave along the time axis.In the context of FM radio signals, the sine function is used to describe the carrier wave, enabling the transmission of signals over long distances. The key to understanding this is realizing that the sine function's shape remains consistent, and these parameters adjust its positioning and size on a graph.Having a firm grasp of the sine function is critical when exploring waves and oscillations, not just in physics, but in several other fields like engineering and even economics.

Frequency

Frequency refers to the number of cycles a wave completes in one second. It is measured in hertz (Hz), where 1 Hz equals one cycle per second. In our FM radio signal example, the frequency is given by the equation \( y = a \, \sin(2 \pi (9.15 \times 10^{7}) t) \). This tells us that the frequency \( f \) is \(9.15 \times 10^{7}\) Hz, which is equivalent to 91.5 MHz.Understanding frequency is crucial because it determines the pitch of a sound or the station on the radio. Higher frequencies result in more wave cycles per period and typically have higher pitches or finer details. When tuning into FM radio, the frequency helps identify different radio stations, which is why each station has a unique frequency assigned to it.

Period of a Wave

The period of a wave is the time it takes to complete one full cycle and is the reciprocal of the frequency. Mathematically, the period \( T \) is expressed as:\[T = \frac{1}{f}\]In our example with the FM radio signal, given that the frequency \( f = 9.15 \times 10^{7} \) Hz, the period \( T \) would be approximately \( 1.093 \times 10^{-8} \) seconds. This means one complete cycle of the FM wave is very brief due to the high frequency.Understanding the period is important as it tells us how short or long it takes between repeated events or cycles. In practical terms, shorter periods (indicating higher frequencies) are used for high-speed communications and data transmissions. The interplay between frequency and period helps engineers and scientists design systems that effectively transmit and receive signals, making it a cornerstone of telecommunications and physics.