To solve this problem, we model the temperature $T$ as a sinusoidal function of time $t$ in hours after midnight. Given that the temperature fluctuates between 78 and 92 degrees, we can start by identifying key characteristics of the function.

Step 1: Determine the amplitude and vertical shift

The temperature varies between a minimum of 78 and a maximum of 92 degrees, so:

• Amplitude $A = \frac{92 - 78}{2} = 7$
• Midline (average temperature) $D = \frac{92 + 78}{2} = 85$

Step 2: Set up the sinusoidal function

The sinusoidal function can be written as: $T(t) = A \cos(B(t - C)) + D$ where:

• $A = 7$
• $D = 85$

Since the average daily temperature first occurs at 9 AM, this point corresponds to the cosine function's "center" (where the cosine function equals 0). Therefore, $C = 9$.

Step 3: Find the period

A full day (24 hours) is the period for this sinusoidal model, so we have: $\text{Period} = \frac{2\pi}{B} = 24$ Solving for $B$: $B = \frac{2\pi}{24} = \frac{\pi}{12}$

Step 4: Write the specific function

Now we can substitute these values into the equation: $T(t) = 7 \cos\left(\frac{\pi}{12}(t - 9)\right) + 85$

Step 5: Solve for $t$ when $T(t) = 82$

We set $T(t) = 82$ and solve for $t$: $82 = 7 \cos\left(\frac{\pi}{12}(t - 9)\right) + 85$ Subtracting 85 from both sides: $-3 = 7 \cos\left(\frac{\pi}{12}(t - 9)\right)$ Dividing by 7: $\cos\left(\frac{\pi}{12}(t - 9)\right) = -\frac{3}{7}$ Now we take the arccosine of both sides: $\frac{\pi}{12}(t - 9) = \cos^{-1}\left(-\frac{3}{7}\right)$ Calculating $\cos^{-1}\left(-\frac{3}{7}\right) \approx 1.99$ radians: $\frac{\pi}{12}(t - 9) = 1.99$ Now, solve for $t$: $t - 9 = \frac{1.99 \times 12}{\pi} \approx 7.60$ $t \approx 7.60 + 9 = 16.60$

Answer

The temperature first reaches 82 degrees approximately 16.60 hours after midnight, which is around 4:36 PM.

Would you like more details on any step, or do you have further questions?

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Here are five related questions to deepen understanding:

1. How would the equation change if the temperature range or timing of the average temperature changes?
2. Can you explain how to find when the temperature next reaches 82 degrees after it first hits that mark?
3. How would this model change if the temperature followed a sine function rather than a cosine function?
4. How do we interpret phase shifts in sinusoidal functions for real-world applications like temperature modeling?
5. What are other real-life situations where sinusoidal models are useful?

Tip: In sinusoidal functions, the period determines how long it takes for the function to complete one cycle. This is particularly useful in periodic phenomena like daily temperatures or seasonal patterns.