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Let's work through the problem step by step, based on the information in the image.

### Problem Breakdown:
- The function \( y(t) \) represents the sound wave intensity in decibels, modeled by a sine wave.
- The peak intensity is 70 dB, the minimum intensity is 50 dB, and the frequency is 2 Hz.

### Step 1: Determine the amplitude, period, and vertical shift of the sound wave function.

1. **Amplitude**:  
   The amplitude \( A \) of the sine wave is half the difference between the peak and minimum values.  
   \[
   A = \frac{\text{peak} - \text{min}}{2} = \frac{70 - 50}{2} = \frac{20}{2} = 10 \, \text{dB}
   \]

2. **Vertical Shift** (also called the "midline" or "DC offset"):  
   The vertical shift \( C \) is the average of the peak and minimum values.  
   \[
   C = \frac{\text{peak} + \text{min}}{2} = \frac{70 + 50}{2} = \frac{120}{2} = 60 \, \text{dB}
   \]

3. **Period** and **Frequency**:  
   The period \( T \) of a sine wave is the reciprocal of the frequency. Since the frequency is given as 2 Hz:  
   \[
   T = \frac{1}{f} = \frac{1}{2} = 0.5 \, \text{seconds}
   \]

The general form of the sine wave is \( y(t) = A \sin(bt) + C \), where \( b \) relates to the period by the formula:
\[
T = \frac{2\pi}{b} \quad \text{or} \quad b = \frac{2\pi}{T}
\]
Substituting \( T = 0.5 \, \text{seconds} \):
\[
b = \frac{2\pi}{0.5} = 4\pi
\]

### Step 2: Write the function \( y(t) \) that describes the intensity of the sound wave.

Using the information gathered:
- Amplitude \( A = 10 \)
- Vertical shift \( C = 60 \)
- \( b = 4\pi \)

The function that models the sound wave is:
\[
y(t) = 10 \sin(4\pi t) + 60
\]

### Step 3: Find the intensity of the sound wave at \( t = 0.75 \) seconds.

Now, substitute \( t = 0.75 \) seconds into the equation:
\[
y(0.75) = 10 \sin(4\pi \times 0.75) + 60
\]
First, calculate \( 4\pi \times 0.75 \):
\[
4\pi \times 0.75 = 3\pi
\]
Since \( \sin(3\pi) = 0 \), the equation simplifies to:
\[
y(0.75) = 10 \times 0 + 60 = 60 \, \text{dB}
\]
So, the intensity at \( t = 0.75 \) seconds is 60 dB.

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### Follow-up Questions:
1. How do we determine the value of \( b \) in the sine function?
2. What is the relationship between period and frequency for a sine wave?
3. How would the function change if the frequency was 3 Hz instead of 2 Hz?
4. Can the amplitude of the wave be negative, and what would that mean physically?
5. What would happen to the function if the peak intensity was 80 dB and the minimum was 60 dB?

### Tip:
When solving trigonometric modeling problems, always pay attention to how amplitude, period, and vertical shifts affect the overall behavior of the sine or cosine function.

A sound wave can be modeled by the function y(t) (in decibels) given by a sine wave. The sound wave has a peak intensity of 70 dB, a minimum intensity of 50 dB, and a frequency of 2 Hz. 1. Determine the amplitude, period, and vertical shift of the sound wave function. 2. Write the function y(t) that describes the intensity of the sound wave. 3. Find the intensity of the sound wave at t = 0.75 seconds.

Learn how to model a sound wave using sine functions with a peak intensity of 70 dB and a minimum intensity of 50 dB. This problem involves determining the amplitude, vertical shift, and frequency to write the function y(t). Suitable for students in Grades 10-12.

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