Let's analyze the problem step by step:

Problem Recap:

• The temperature follows a sinusoidal pattern modeled by a sine function $f(x)$.
• Maximum temperature: $120^\circ$ at $t = 14 \, \text{minutes}$.
• Minimum temperature: $96^\circ$, which occurs $28 \, \text{minutes}$ after the maximum.

Step 1: Determine the midline of the sine wave

The midline is the average of the maximum and minimum temperatures: $\text{Midline} = \frac{\text{Maximum Temperature} + \text{Minimum Temperature}}{2} = \frac{120 + 96}{2} = 108$ So, the midline of the sine wave is at $y = 108$.

Step 2: Determine the amplitude

The amplitude is the difference between the maximum temperature and the midline: $\text{Amplitude} = \text{Maximum Temperature} - \text{Midline} = 120 - 108 = 12$

Step 3: Determine the period

The problem states that the minimum temperature occurs 28 minutes after the maximum temperature. Since a full period spans one complete oscillation (from max to min to max), the total period is: $\text{Period} = 2 \times 28 = 56 \, \text{minutes}$

The formula for the period of a sine wave is: $\text{Period} = \frac{2\pi}{b}$ Solving for $b$: $b = \frac{2\pi}{\text{Period}} = \frac{2\pi}{56} = \frac{\pi}{28}$

Step 4: Phase shift

The sine function normally starts at $0$, but the maximum occurs at $t = 14$. Therefore, the phase shift is $14$. Since the sine function is shifted to the right, this corresponds to a positive phase shift.

Step 5: Write the equation

The general form of a sine function is: $f(x) = a \sin(b(x - c)) + d$ Where:

• $a = 12$ (amplitude),
• $b = \frac{\pi}{28}$ (frequency),
• $c = 14$ (phase shift),
• $d = 108$ (midline).

Thus, the equation is: $f(x) = 12 \sin\left(\frac{\pi}{28}(x - 14)\right) + 108$

Would you like a detailed explanation for any specific step? Here are five follow-up questions you could explore:

1. How do we know the period is $2 \times 28$ instead of just $28$?
2. What happens if we use a cosine function instead of sine for this scenario?
3. How does the phase shift affect the graph of the function?
4. Can we determine the function using the minimum instead of the maximum point?
5. How do we verify this equation is correct by substituting values?

Tip: Always confirm the midline and amplitude by checking their symmetry between the max and min values!