To find the equation for the temperature $D(t)$ as a sinusoidal function of time $t$, let's analyze the information given:

1. Midline (average temperature):

   • The high temperature during the day is 70°F, and the low is 40°F.
   • The average temperature (midline) is: $\frac{70 + 40}{2} = 55$
   • This value, 55°F, represents the centerline or the vertical shift of the sinusoidal function.

2. Amplitude:

   • The amplitude is the difference between the average temperature and the high (or low) temperature: $70 - 55 = 15$

3. Period:

   • Since this is a daily cycle, the temperature pattern repeats every 24 hours. Therefore, the period $T$ is 24 hours.
   • For a sinusoidal function $\cos$ or $\sin$, the period $T$ is given by $T = \frac{2\pi}{B}$, so we solve for $B$: $B = \frac{2\pi}{24} = \frac{\pi}{12}$

4. Phase Shift:

   • Since the temperature is 55°F at midnight (which we can take as $t = 0$), we can use a cosine function starting at the midline at $t = 0$.
   • This makes the cosine function fit without a phase shift.

With these parameters, the equation for $D(t)$ becomes:

$D(t) = 55 + 15 \cos\left(\frac{\pi}{12} t\right)$

Final Answer:

$D(t) = 55 + 15 \cos\left(\frac{\pi}{12} t\right)$

Would you like further details on any of these steps?

Here are some related questions you might find useful:

1. What would the equation look like if the high temperature occurred at a different time of day?
2. How would the equation change if the period were 12 hours instead of 24 hours?
3. What is the temperature at noon according to this model?
4. How can we use a sine function instead of a cosine function for this model?
5. How would the amplitude change if the high and low temperatures were closer together?

Tip: In sinusoidal models, always identify the midline, amplitude, period, and phase shift for accuracy.