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Let’s solve this step by step. The problem describes the temperature over a day modeled as a sinusoidal function.

### Key Information:
1. **High temperature**: 53°F  
2. **Low temperature**: 37°F  
3. **Temperature at midnight (t = 0)**: 45°F  
4. **Time period**: 24 hours (since temperature cycles daily)  
5. \( t \): Number of hours since midnight.

---

### Steps to Create the Equation:

#### 1. Determine the **midline** of the sinusoidal function:
The midline is the average of the high and low temperatures:
\[
\text{Midline} = \frac{\text{High} + \text{Low}}{2} = \frac{53 + 37}{2} = 45
\]

#### 2. Determine the **amplitude**:
The amplitude is the difference between the high (or low) temperature and the midline:
\[
\text{Amplitude} = \text{High} - \text{Midline} = 53 - 45 = 8
\]

#### 3. Determine the **period**:
The period of the sine or cosine function is the time it takes to complete one full cycle. Since the temperature cycles daily (24 hours):
\[
\text{Period} = 24 \quad \Rightarrow \quad \text{B} = \frac{2\pi}{\text{Period}} = \frac{2\pi}{24} = \frac{\pi}{12}
\]

#### 4. Choose a sinusoidal function:
- Since the temperature starts at 45°F at midnight (which is the midline), a **cosine function** works well, as cosine starts at its peak or midline.

The general cosine function is:
\[
D(t) = A \cos(Bt - C) + D
\]
Where:
- \( A \): Amplitude (\( 8 \))  
- \( B \): Frequency (\( \frac{\pi}{12} \))  
- \( C \): Phase shift (\( 0 \), since it starts at \( t = 0 \))  
- \( D \): Midline (\( 45 \))

---

#### 5. Write the final equation:
\[
D(t) = 8 \cos\left(\frac{\pi}{12}t\right) + 45
\]

This equation models the temperature \( D(t) \) as a function of time \( t \).

---

Would you like further clarification or a graph of the function?  

### Related Questions:
1. How does the equation change if the temperature is at its maximum at \( t = 6 \) instead of midnight?
2. What is the significance of the phase shift \( C \) in sinusoidal equations?
3. How would the model change if the period were 12 hours instead of 24 hours?
4. How can you verify the equation by substituting known values of \( t \)?
5. What is the derivative of \( D(t) \), and how does it relate to temperature changes?

### Tip:
When modeling periodic phenomena, always double-check the amplitude, period, and midline values to ensure they align with the real-world context.

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 45 degrees at midnight, and the high and low temperatures during the day are 53 and 37 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D(t), in terms of t.

Learn how to model temperature over a day as a sinusoidal function. Given a high of 53°F, a low of 37°F, and 45°F at midnight, find the temperature equation D(t) using trigonometric principles. Perfect for high school trigonometry students.

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