Sure, let's walk through this step-by-step!
### Step 1: Understanding the Wave Equation
The given sound wave is defined by the equation:
[tex]\[
y = 6 \sin (324 \pi t)
\][/tex]
This is a sine function which represents a sound wave where:
- The amplitude (maximum value of [tex]\( y \)[/tex]) is [tex]\( 6 \)[/tex].
- The angular frequency [tex]\( \omega \)[/tex] is [tex]\( 324 \pi \)[/tex].
### Step 2: Finding the Angular Frequency
The general form of a sine wave is:
[tex]\[
y = A \sin(\omega t)
\][/tex]
Here, [tex]\(\omega\)[/tex] (angular frequency) is given as [tex]\( 324 \pi \)[/tex].
### Step 3: Converting Angular Frequency to Regular Frequency
Angular frequency [tex]\(\omega\)[/tex] is related to the regular frequency [tex]\( f \)[/tex] through the formula:
[tex]\[
\omega = 2 \pi f
\][/tex]
Given [tex]\(\omega = 324 \pi\)[/tex], we can solve for the frequency [tex]\( f \)[/tex]:
[tex]\[
324 \pi = 2 \pi f
\][/tex]
Dividing both sides by [tex]\( 2 \pi \)[/tex]:
[tex]\[
f = \frac{324 \pi}{2 \pi} = 162 \text{ Hz}
\][/tex]
So, the frequency [tex]\( f \)[/tex] is [tex]\( 162 \)[/tex] cycles per second.
### Step 4: Determining the Time Interval
We are given the times [tex]\( t = 3 \)[/tex] seconds and [tex]\( t = 5.5 \)[/tex] seconds. The time interval ([tex]\(\Delta t\)[/tex]) between these two points is:
[tex]\[
\Delta t = t_{\text{end}} - t_{\text{start}} = 5.5 - 3 = 2.5 \text{ seconds}
\][/tex]
### Step 5: Calculating the Number of Cycles
The number of cycles that occur within a given time interval can be found by multiplying the frequency by the time interval:
[tex]\[
\text{Number of cycles} = \text{Frequency} \times \Delta t
\][/tex]
Substitute the values:
[tex]\[
\text{Number of cycles} = 162 \, \text{Hz} \times 2.5 \, \text{seconds} = 405 \, \text{cycles}
\][/tex]
### Final Answer
The number of cycles of the sound wave occurring between [tex]\( t = 3 \)[/tex] seconds and [tex]\( t = 5.5 \)[/tex] seconds is:
[tex]\[
405 \, \text{cycles}
\][/tex]
