This problem involves modeling temperature over a day as a sinusoidal function. Let's break it down step by step to construct the equation for temperature $D$ as a function of time $t$ (in hours since midnight):

Step 1: Identify Key Information

1. Midline temperature: The temperature at midnight is $75^\circ$, and the temperature oscillates around this value. This is the average of the high and low temperatures: $\text{Midline} = \frac{\text{High} + \text{Low}}{2} = \frac{95 + 55}{2} = 75.$ So, $75$ is the vertical shift of the sinusoidal function.

2. Amplitude: The amplitude is half the difference between the high and low temperatures: $\text{Amplitude} = \frac{\text{High} - \text{Low}}{2} = \frac{95 - 55}{2} = 20.$ This represents how far the temperature oscillates above and below the midline.

3. Period: A full cycle of temperature variation occurs over 24 hours. The general period formula for a sinusoidal function is: $\text{Period} = \frac{2\pi}{B},$ where $B$ is the frequency. Setting the period to $24$: $24 = \frac{2\pi}{B} \implies B = \frac{2\pi}{24} = \frac{\pi}{12}.$

4. Phase shift: At $t = 0$ (midnight), the temperature is $75^\circ$, which is the midline value. This indicates a cosine function, as $\cos(0) = 1$. No horizontal shift is needed.

Step 2: Write the Equation

The general form of a sinusoidal function is: $D(t) = A \cos(Bt + C) + D.$

• $A = 20$ (amplitude),
• $B = \frac{\pi}{12}$ (frequency),
• $C = 0$ (no phase shift),
• $D = 75$ (midline).

Thus, the equation becomes: $D(t) = 20 \cos\left(\frac{\pi}{12} t\right) + 75.$

Final Answer

The temperature $D$ as a function of time $t$ is: $\boxed{D(t) = 20 \cos\left(\frac{\pi}{12} t\right) + 75.}$

Would you like a detailed explanation of the graph or any specific part of this derivation?

Here are 5 follow-up questions to consider:

1. How would the equation change if the temperature were modeled using a sine function instead of cosine?
2. Can you determine the temperature at $t = 6$ hours (6:00 AM) using this equation?
3. How would the equation adjust if the high and low temperatures changed?
4. What does the frequency $\frac{\pi}{12}$ represent in practical terms?
5. Can you verify that this function correctly models the temperature at both the high and low points?

Tip: The cosine function is often used when a cycle starts at a maximum or midline value, while sine is used for cycles starting at a minimum.