To model the temperature with a sinusoidal function, we need to use the general form of the sinusoidal function:

$D(t) = A \sin(B(t - C)) + D$

Where:

• $A$ is the amplitude (half the difference between the high and low temperatures),
• $B$ controls the period (how long it takes for the temperature to repeat a cycle),
• $C$ is the horizontal shift (where the maximum or minimum temperature occurs),
• $D$ is the vertical shift (the average temperature).

Given the information:

• The low temperature is 59°F, the high is 83°F, and the temperature at midnight (t = 0) is 71°F.
• The period is 24 hours (since a full cycle is one day).

Step 1: Find the amplitude $A$

The amplitude is half the difference between the high and low temperatures:

$A = \frac{83 - 59}{2} = 12$

Step 2: Find the vertical shift $D$

The vertical shift is the average of the high and low temperatures:

$D = \frac{83 + 59}{2} = 71$

Step 3: Find the period and $B$

The period is 24 hours, and since the formula for the period of a sine function is:

$\text{Period} = \frac{2\pi}{B}$

We can solve for $B$:

$24 = \frac{2\pi}{B} \quad \Rightarrow \quad B = \frac{\pi}{12}$

Step 4: Find the horizontal shift $C$

At midnight (t = 0), the temperature is 71°F, which is the average temperature. Since this occurs at $t = 0$, the sine function must start at its equilibrium position, which suggests no horizontal shift. Thus:

$C = 0$

Final equation:

$D(t) = 12 \sin\left(\frac{\pi}{12}t\right) + 71$

This is the equation that models the temperature as a sinusoidal function of time, $t$, in hours since midnight.

Would you like more details on any of the steps? Here are some related questions:

1. How does the amplitude of the sinusoidal function affect the temperature variation?
2. How would the equation change if the temperature reached a maximum at a different time of day?
3. How can we adjust the equation if the period of the temperature cycle was longer than 24 hours?
4. What does the vertical shift represent in terms of the real-world temperature?
5. How could we apply this model to predict temperatures for multiple days?

Tip: Always carefully check your given data and ensure you are using the correct values for shifts, amplitude, and period to build an accurate sinusoidal model.