To model the temperature as a sinusoidal function, we use the general equation of a sinusoidal function:

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

Step-by-Step Process:

1. Amplitude ($A$): The amplitude is half the difference between the maximum and minimum temperatures: $A = \frac{\text{High} - \text{Low}}{2} = \frac{95 - 59}{2} = 18$

2. Midline ($D$): The midline is the average of the maximum and minimum temperatures: $D = \frac{\text{High} + \text{Low}}{2} = \frac{95 + 59}{2} = 77$

3. Period ($P$) and $B$: Since the temperature completes one full cycle in 24 hours, the period is 24. The relationship between $P$ and $B$ is: $P = \frac{2\pi}{B} \implies B = \frac{2\pi}{P} = \frac{2\pi}{24} = \frac{\pi}{12}$

4. Phase Shift ($C$): At $t = 0$ (midnight), the temperature is at its maximum ($95$), which corresponds to a cosine function starting at its peak. However, since we are using a sine function, we introduce a phase shift to align it:

   • A sine function starts at the midline. To reach the maximum at $t = 0$, the sine function needs to shift $\frac{P}{4} = \frac{24}{4} = 6$ hours to the left. Therefore, $C = -6$.

   Alternatively, you could rewrite this as $(t + 6)$ for simplicity.

Final Equation:

Substituting all values: $D(t) = 18 \sin\left(\frac{\pi}{12}(t + 6)\right) + 77$

This equation models the temperature as a sinusoidal function.

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Would you like further clarification on any part of this derivation? Let me know if you have questions!

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5 Related Questions:

1. How can we interpret the amplitude and midline in real-world temperature models?
2. What would the equation look like if the temperature peaked at a different time?
3. How does changing the period of the function affect the temperature model?
4. Can this equation be rewritten using a cosine function? How?
5. What other natural phenomena can be modeled using sinusoidal functions?

Tip:

When modeling with sinusoidal functions, always start by identifying the amplitude, midline, and period before choosing the appropriate phase shift.