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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?

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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.