At A Certain Location, The Number Of Hours Of Sunlight Is Modeled By Y = 6.4 Cos ⁡ ( Π 26 X ) + 12 Y=6.4 \cos \left(\frac{\pi}{26} X\right)+12 Y = 6.4 Cos ( 26 Π ​ X ) + 12 , Where X X X Represents The Number Of Weeks After The Summer Solstice.Based On The Model, What Is The Minimum Number Of

The given model, $y=6.4 \cos \left(\frac{\pi}{26} x\right)+12$, represents the number of hours of sunlight at a certain location, where $x$ is the number of weeks after the summer solstice. This model is based on a cosine function, which is a periodic function that oscillates between a maximum and minimum value. In this case, the model oscillates between a minimum of 12 hours and a maximum of 18 hours of sunlight.

The Cosine Function

The cosine function is a fundamental function in mathematics that is used to model periodic phenomena. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the context of the given model, the cosine function is used to represent the number of hours of sunlight as a function of the number of weeks after the summer solstice.

Key Components of the Model

The given model has several key components that are worth noting:

• Amplitude: The amplitude of the model is 6.4, which represents the maximum deviation from the average value of 12 hours of sunlight.
• Period: The period of the model is 26 weeks, which represents the time it takes for the model to complete one full cycle.
• Phase Shift: The phase shift of the model is 0, which means that the model starts at its maximum value at the summer solstice.

Finding the Minimum Number of Hours of Sunlight

To find the minimum number of hours of sunlight, we need to find the value of $y$ when $x$ is at its maximum value. Since the model is a cosine function, the maximum value of $x$ is 26 weeks. Substituting $x=26$ into the model, we get:

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

Simplifying the expression, we get:

$$y=6.4 \cos (\pi)+12$$

Since $\cos (\pi)=-1$, we have:

$$y=6.4 \cdot (-1)+12$$

$$y=-6.4+12$$

$$y=5.6$$

Therefore, the minimum number of hours of sunlight is 5.6 hours.

Interpretation of the Results

The results of this analysis can be interpreted in several ways:

• Minimum Sunlight Hours: The minimum number of hours of sunlight is 5.6 hours, which occurs 26 weeks after the summer solstice.
• Sunlight Patterns: The model shows that the number of hours of sunlight follows a periodic pattern, with a maximum of 18 hours and a minimum of 5.6 hours.
• Seasonal Variations: The model highlights the seasonal variations in sunlight hours, with more hours of sunlight during the summer months and fewer hours during the winter months.

Conclusion

Q: What is the purpose of the given model?

A: The given model, $y=6.4 \cos \left(\frac{\pi}{26} x\right)+12$, is used to represent the number of hours of sunlight at a certain location, where $x$ is the number of weeks after the summer solstice.

Q: What is the amplitude of the model?

A: The amplitude of the model is 6.4, which represents the maximum deviation from the average value of 12 hours of sunlight.

Q: What is the period of the model?

A: The period of the model is 26 weeks, which represents the time it takes for the model to complete one full cycle.

Q: What is the phase shift of the model?

A: The phase shift of the model is 0, which means that the model starts at its maximum value at the summer solstice.

Q: How do I find the minimum number of hours of sunlight?

A: To find the minimum number of hours of sunlight, you need to find the value of $y$ when $x$ is at its maximum value. Since the model is a cosine function, the maximum value of $x$ is 26 weeks. Substituting $x=26$ into the model, you can find the minimum number of hours of sunlight.

Q: What is the minimum number of hours of sunlight?

A: The minimum number of hours of sunlight is 5.6 hours, which occurs 26 weeks after the summer solstice.

Q: How does the model show seasonal variations?

A: The model shows that the number of hours of sunlight follows a periodic pattern, with a maximum of 18 hours and a minimum of 5.6 hours. This highlights the seasonal variations in sunlight hours, with more hours of sunlight during the summer months and fewer hours during the winter months.

Q: Can I use this model to predict sunlight hours for other locations?

A: No, this model is specific to a certain location and cannot be used to predict sunlight hours for other locations. The model is based on the specific conditions of the location, including the latitude and longitude.

Q: How accurate is the model?

A: The accuracy of the model depends on the data used to create it. If the data is accurate and reliable, the model will be more accurate. However, if the data is incomplete or inaccurate, the model may not be as accurate.

Q: Can I modify the model to fit my specific needs?

A: Yes, you can modify the model to fit your specific needs. However, you will need to have a good understanding of the underlying mathematics and the data used to create the model.

Q: Where can I find more information about the model?

A: You can find more information about the model by consulting the original source or by searching online for related resources.