Problem 24 - Algebra 2

Question

MEDICINE For Exercises 24 and \(25,\) use the following information. If the amplitude of the sine function is \(0.25,\) write the equations for tuning forks that resonate with a frequency of \(64,256,\) and 512 Hertz.

Step-by-Step Solution

Answer

Equations: \( y = 0.25 \sin(128\pi t) \), \( y = 0.25 \sin(512\pi t) \), \( y = 0.25 \sin(1024\pi t) \).

Step 1: Understand Amplitude and Frequency

The amplitude of a sine function determines the maximum value it reaches from its equilibrium, and for all frequencies, it is given as 0.25. The frequency determines how many cycles occur in a unit of time. Here, you need the equations for three different frequencies: 64 Hz, 256 Hz, and 512 Hz.

Step 2: Write the Generic Sine Function Equation

The general equation for a sine function can be written as \[ y = A \sin(2\pi f t) \]where \( A \) is the amplitude and \( f \) is the frequency.

Step 3: Plug in the Values for 64 Hz

Substitute the given amplitude, 0.25, and frequency, 64 Hz, into the equation:\[ y = 0.25 \sin(2\pi \times 64 \times t) \] This is the equation for a tuning fork with a frequency of 64 Hz.

Step 4: Plug in the Values for 256 Hz

Using the same approach, substitute the amplitude and 256 Hz into the equation:\[ y = 0.25 \sin(2\pi \times 256 \times t) \] This is the equation for a tuning fork with a frequency of 256 Hz.

Step 5: Plug in the Values for 512 Hz

Finally, substitute the amplitude and 512 Hz into the equation:\[ y = 0.25 \sin(2\pi \times 512 \times t) \] This is the equation for a tuning fork with a frequency of 512 Hz.

Key Concepts

Understanding Amplitude in a Sine Function

Amplitude is a fundamental concept when dealing with sine functions. It signifies how far the wave stretches from its central axis or equilibrium position. Think of it like the height of ocean waves. In mathematics, the amplitude is represented by a number that determines the peak of the wave compared to its midpoint.
For example:

• If an amplitude is larger, the wave reaches higher and lower extremes.
• If an amplitude is smaller, the wave appears more compressed and shorter.

This maximum deviation is crucial as it dictates how intense the oscillations of the wave are. In many real-world applications like sound waves or heartbeats, amplitude helps us understand intensity or volume. If you see an equation with amplitude 0.25, it means that the sine wave will vary from 0.25 to -0.25 across its cycle.

Decoding Frequency in Sine Wave Equations

Frequency is another key factor in sine functions and tells us how frequently the wave cycles through its pattern in a given time period.

• The frequency is measured in Hertz (Hz), which is equivalent to the number of cycles per second.
• A higher frequency implies the wave cycles more times in one second, making the wave look more compressed on a graph.
• A lower frequency results in fewer wave cycles per second, making the graph appear more stretched out.

Frequency is vital in applications such as tuning instruments, where the precise frequency determines the pitch of the sound. In our example, frequencies such as 64 Hz, 256 Hz, or 512 Hz mean the wave completes that many full cycles in a second. When inserting these frequencies into our sine equation, we adjust how the wave pattern progresses over time, depicting different sound pitches.

Writing and Solving Trigonometric Equations

Trigonometric equations involving sine functions are often used to model periodic phenomena. To construct these equations, it's helpful to use the general sine wave formula: \[ y = A \sin(2\pi f t) \]

• A is the amplitude, showing the wave's maximum travel distance from its central position.
• f represents the frequency, determining the speed at which cycles occur within a time frame.

For example, when constructing a trigonometric equation for a tuning fork:

• If the amplitude is 0.25 and the frequency is 64 Hz, the equation is \( y = 0.25 \sin(2\pi \times 64 \times t) \).
• For a frequency of 256 Hz with the same amplitude, the equation becomes \( y = 0.25 \sin(2\pi \times 256 \times t) \).
• At 512 Hz, the expression becomes \( y = 0.25 \sin(2\pi \times 512 \times t) \).

By solving these equations, we can simulate the respective sound waves produced by each tuning fork, which helps us analyze and understand various harmonic patterns.