Outside temperatures over the course of one full day can be modeled as a sinusoidal function. Suppose the temperature varies between  
69
 °F and  
83
 °F during the day and the average daily temperature first occurs at  
12
  AM. The temperature continues to decrease until the sun rises. How many hours after midnight does the temperature first reach  
74
 °F?

Write the sinusoidal function modeling the temperature starting at  
12
  AM in terms of  
x
 :

y
=
 
  

How many hours after midnight does the temperature first reach  
74
 °F?

(Solve by graphing and round your answer to the nearest hundredth)
x
≈
 
 hours

### Step 1: Model the temperature as a sinusoidal function

We know that the temperature fluctuates between 69°F and 83°F over the course of the day. The general form of a sinusoidal function modeling temperature is:

\[
y = A \cdot \sin(B(x - C)) + D
\]

Where:
- \( A \) is the amplitude,
- \( B \) is related to the period,
- \( C \) is the horizontal shift (phase shift),
- \( D \) is the vertical shift (average value).

#### Determine the parameters:
- **Amplitude** \( A \): The amplitude is half the distance between the maximum and minimum temperatures:
  \[
  A = \frac{\text{max} - \text{min}}{2} = \frac{83 - 69}{2} = 7
  \]

- **Vertical shift** \( D \): This is the average temperature, which is halfway between the maximum and minimum temperatures:
  \[
  D = \frac{\text{max} + \text{min}}{2} = \frac{83 + 69}{2} = 76
  \]

- **Period**: The temperature fluctuates over a full day (24 hours), so the period is 24 hours. The relationship between the period and \( B \) is:
  \[
  \text{Period} = \frac{2\pi}{B} \quad \text{so} \quad B = \frac{2\pi}{24} = \frac{\pi}{12}
  \]

- **Phase shift** \( C \): Since the average daily temperature first occurs at 12 AM, we can assume that there is no horizontal shift, so \( C = 0 \).

Thus, the sinusoidal function becomes:
\[
y = 7 \cdot \sin\left(\frac{\pi}{12}x\right) + 76
\]

### Step 2: Find when the temperature first reaches 74°F

We need to solve for \( x \) when the temperature \( y = 74 \)°F. Set the equation equal to 74 and solve for \( x \):

\[
74 = 7 \cdot \sin\left(\frac{\pi}{12}x\right) + 76
\]

Subtract 76 from both sides:

\[
-2 = 7 \cdot \sin\left(\frac{\pi}{12}x\right)
\]

Divide by 7:

\[
\sin\left(\frac{\pi}{12}x\right) = -\frac{2}{7}
\]

Now, take the inverse sine of both sides:

\[
\frac{\pi}{12}x = \arcsin\left(-\frac{2}{7}\right)
\]

Using a calculator to find \( \arcsin\left(-\frac{2}{7}\right) \):

\[
\frac{\pi}{12}x \approx -0.29146
\]

Solve for \( x \):

\[
x \approx \frac{-0.29146 \cdot 12}{\pi} \approx -1.11
\]

This value represents the first time the temperature reaches 74°F, but it's a negative value, which corresponds to a time before midnight. To find the next instance when the temperature reaches 74°F, we use the fact that the sine function has periodic behavior.

The next solution within the first 24 hours can be found using the symmetry of the sine wave. Add \( \pi \) to account for the next intersection:

\[
\frac{\pi}{12}x = \pi - 0.29146
\]

\[
\frac{\pi}{12}x = 2.85013
\]

Solve for \( x \):

\[
x \approx \frac{2.85013 \cdot 12}{\pi} \approx 10.88
\]

Thus, the temperature first reaches 74°F approximately **10.88 hours after midnight**.

### Final Answer:
The temperature first reaches 74°F approximately 10.88 hours after midnight, which is around 10:53 AM.

Would you like any further details or clarifications?

---

**Related questions:**
1. How would the equation change if the minimum temperature occurred at 6 AM instead of 12 AM?
2. What is the temperature 8 hours after midnight based on this model?
3. How many times per day does the temperature reach 74°F?
4. How would the function change if the maximum temperature occurred at 2 PM?
5. What is the period of the sinusoidal function in terms of hours?

**Tip:** The amplitude of a sinusoidal function always represents half the distance between the maximum and minimum values of the function.

Outside temperatures over the course of one full day can be modeled as a sinusoidal function. Suppose the temperature varies between 69°F and 83°F during the day, and the average daily temperature first occurs at 12 AM. The temperature continues to decrease until the sun rises. How many hours after midnight does the temperature first reach 74°F? Write the sinusoidal function modeling the temperature starting at 12 AM in terms of x: y = ? How many hours after midnight does the temperature first reach 74°F?

This problem models daily temperature variations using a sinusoidal function. Given a temperature range from 69°F to 83°F and an average temperature at 12 AM, learn how to calculate the time when the temperature first reaches 74°F, using key trigonometric concepts and formulas. Suitable for Grades 10-12.

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