We need to find a sinusoidal function of the form:

$y = A \sin(\omega x - \phi) + B$

where:

• The maximum number of daylight hours is 15.788 (on the summer solstice, $x = 172$).
• The minimum number of daylight hours is 8.276 (on the winter solstice, $x \approx 355$).
• The period is 365 days (one full cycle from summer solstice to the next summer solstice).

Step 1: Find the Amplitude $A$

The amplitude of a sinusoidal function is given by:

$A = \frac{\text{max} - \text{min}}{2}$

$A = \frac{15.788 - 8.276}{2} = \frac{7.512}{2} = 3.756$

Step 2: Find the Midline $B$

The midline (vertical shift) is the average of the maximum and minimum values:

$B = \frac{\text{max} + \text{min}}{2}$

$B = \frac{15.788 + 8.276}{2} = \frac{24.064}{2} = 12.032$

Step 3: Find the Angular Frequency $\omega$

The period of the function is 365 days, so we use the standard formula:

$\omega = \frac{2\pi}{\text{period}}$

$\omega = \frac{2\pi}{365}$

Step 4: Find the Phase Shift $\phi$

Since the function should reach its maximum (15.788 hours) at $x = 172$, we set up the equation:

$\sin(\omega \cdot 172 - \phi) = 1$

Since $\sin$ reaches its maximum at $\frac{\pi}{2}$, we set:

$\omega \cdot 172 - \phi = \frac{\pi}{2}$

Substituting $\omega = \frac{2\pi}{365}$:

$\frac{2\pi}{365} \cdot 172 - \phi = \frac{\pi}{2}$

$\frac{344\pi}{365} - \phi = \frac{\pi}{2}$

Solving for $\phi$:

$\phi = \frac{344\pi}{365} - \frac{\pi}{2}$

$\phi = \frac{344\pi}{365} - \frac{182.5\pi}{365}$

$\phi = \frac{161.5\pi}{365}$

Final Equation

$y = 3.756 \sin\left(\frac{2\pi}{365} x - \frac{161.5\pi}{365}\right) + 12.032$

Would you like a breakdown of any specific step?

Related Questions

1. How do you determine the period of a sinusoidal function?
2. Why is the sine function used instead of cosine in this problem?
3. How would the function change if the number of daylight hours varied in a different pattern?
4. Can we use a cosine function instead of a sine function for this problem?
5. How would the function change if the period were 366 days (leap year)?

Tip

A phase shift helps align a sinusoidal function to real-world data, ensuring that the peaks and troughs occur at the correct times.