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.