The given model for the number of hours of sunlight is:

$y = 6.4 \cdot \cos\left(\frac{\pi}{26}x\right) + 12$

where $x$ represents the number of weeks after the summer solstice.

To determine the minimum number of hours of sunlight, let's examine the cosine function, which oscillates between -1 and 1. The general form of a cosine function is:

$y = A \cdot \cos(Bx) + C$

In this case:

• $A = 6.4$, which controls the amplitude (how much the function oscillates),
• $B = \frac{\pi}{26}$, which controls the period (how fast the function oscillates),
• $C = 12$, which shifts the function vertically.

The cosine function reaches its minimum value of $-1$. Substituting this into the equation:

$y_{\text{min}} = 6.4 \cdot (-1) + 12 = -6.4 + 12 = 5.6$

Thus, the minimum number of hours of sunlight at this location is 5.6 hours.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the maximum number of hours of sunlight based on this model?
2. How would the equation change if the amplitude of sunlight variation increased?
3. What is the period of this function, and how many weeks does it represent?
4. What is the value of $y$ when $x = 0$ (i.e., at the summer solstice)?
5. How does changing the value of $C$ affect the model?

Tip: The vertical shift $C$ in trigonometric functions directly affects the average value of the function, influencing the midline between the maximum and minimum values.