Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 73 and 97 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 82 degrees?


To model the temperature as a sinusoidal function, we can start with the general form of a sinusoidal function:

\[
T(t) = A \sin(B(t - C)) + D
\]

where:
- \(A\) is the amplitude,
- \(B\) relates to the period,
- \(C\) is the phase shift,
- \(D\) is the vertical shift.

### Step 1: Determine the Parameters

1. **Amplitude (\(A\))**:
   The temperature varies between 73 and 97 degrees. Thus, the amplitude is half the difference between the maximum and minimum temperatures:
   \[
   A = \frac{97 - 73}{2} = 12
   \]

2. **Vertical Shift (\(D\))**:
   The average temperature \(D\) is the midpoint of the maximum and minimum:
   \[
   D = \frac{73 + 97}{2} = 85
   \]

3. **Period (\(B\))**:
   The temperature cycle completes in 24 hours, so the period is 24 hours. The formula for \(B\) is given by:
   \[
   B = \frac{2\pi}{\text{period}} = \frac{2\pi}{24} = \frac{\pi}{12}
   \]

4. **Phase Shift (\(C\))**:
   The average temperature occurs at 8 AM (8 hours after midnight). Since the sinusoidal function starts at its midpoint when \(t = 0\), we need to adjust for the phase shift. The cosine function starts at its maximum, so we can write:
   \[
   T(t) = 12 \cos\left(\frac{\pi}{12}(t - 8)\right) + 85
   \]

### Step 2: Set Up the Equation for 82 Degrees

We want to find \(t\) when the temperature \(T(t) = 82\):
\[
82 = 12 \cos\left(\frac{\pi}{12}(t - 8)\right) + 85
\]

### Step 3: Solve for \(t\)

1. Rearrange the equation:
   \[
   12 \cos\left(\frac{\pi}{12}(t - 8)\right) = 82 - 85
   \]
   \[
   12 \cos\left(\frac{\pi}{12}(t - 8)\right) = -3
   \]
   \[
   \cos\left(\frac{\pi}{12}(t - 8)\right) = -\frac{1}{4}
   \]

2. Solve for \(\frac{\pi}{12}(t - 8)\):
   \[
   \frac{\pi}{12}(t - 8) = \cos^{-1}\left(-\frac{1}{4}\right)
   \]
   The cosine inverse gives two angles in the range [0, 2π]:
   \[
   \theta_1 = \cos^{-1}\left(-\frac{1}{4}\right) \quad \text{and} \quad \theta_2 = 2\pi - \cos^{-1}\left(-\frac{1}{4}\right)
   \]

3. Calculate \(t\):
   \[
   t_1 = 12 \cdot \frac{\theta_1}{\pi} + 8 \quad \text{and} \quad t_2 = 12 \cdot \frac{\theta_2}{\pi} + 8
   \]

4. Substitute to find the specific values for \(t_1\) and \(t_2\).

### Step 4: Numerical Calculation

Using a calculator, we find:

\[
\theta_1 \approx 1.82 \quad \text{radians} \quad (\text{for } -\frac{1}{4})
\]

Now calculating \(t_1\):
\[
t_1 = 12 \cdot \frac{1.82}{\pi} + 8 \approx 12 \cdot 0.58 + 8 \approx 6.96 + 8 \approx 14.96 \text{ hours}
\]

Now for \(t_2\):
\[
\theta_2 \approx 4.46 \text{ radians}
\]
\[
t_2 = 12 \cdot \frac{4.46}{\pi} + 8 \approx 12 \cdot 1.42 + 8 \approx 17.06 + 8 \approx 25.06 \text{ hours} \quad (\text{not applicable as it's past 24 hours})
\]

### Conclusion

Thus, the first occurrence of 82 degrees occurs at approximately:

\[
\boxed{14.96} \text{ hours after midnight}
\]

### Do you want more details or have any questions? 

Here are some follow-up questions you might consider:
1. How does changing the temperature range affect the sinusoidal model?
2. What would happen if the average temperature shifted to a different time?
3. Can this model be applied to other daily temperature patterns?
4. How would you graph this function over a 24-hour period?
5. What other factors could affect daily temperature variations? 

**Tip**: Always verify the phase shift and amplitude when dealing with sinusoidal functions to ensure accurate modeling.

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 73 and 97 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 82 degrees?

Discover how to model daily temperature variations using sinusoidal functions. This problem illustrates the calculation of when the temperature first reaches 82 degrees, given a range of 73 to 97 degrees and an average temperature at 8 AM. Detailed step-by-step solution provided.

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