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Modeling daylight hours involves creating a mathematical representation of how many hours of daylight occur on a given day.
This can be useful for understanding predictably changing patterns throughout the year.
For the period from April 22 to August 22, we see a gradual increase in daylight hours moving toward the summer solstice, and a gradual decrease after it.
To accurately represent this phenomenon, we need a function that captures not just the change in daylight hours but also the varying rate at which it changes.
This makes a quadratic function suitable because it can model the non-linear changes in daylight over this period.
In simple terms, as we get closer to the summer solstice, the rate of increase in daylight hours slows down and then starts to decrease after the solstice passes.
This non-constant rate of change needs to be mathematically represented using quadratic equations.

The rate of change refers to how quickly or slowly a particular quantity increases or decreases over time.
For daylight hours, this rate is not constant.
As we approach the summer solstice, the rate at which daylight increases begins to slow down, reaches zero at the peak, and then the rate becomes negative as daylight starts decreasing.
This varying rate of change means that a simple straight-line (linear) model won't do the job as it assumes a constant rate of change.
Therefore, a quadratic function is preferred.
The quadratic function allows us to show how daylight hours gradually increase, level off, and then decrease in a smooth curve.
This is more reflective of real-world daylight patterns around the solstice.
Mathematically, if we were to plot this function, we'd see a parabola with its vertex representing the solstice where the rate of change is zero.

The summer solstice marks the day with the longest period of daylight in a year.
It usually occurs around June 21st in the Northern Hemisphere.
Leading up to this day, daylight hours increase.
After the solstice, daylight hours begin to decrease.
This point is significant in daylight hour modeling because it represents a peak or vertex in our quadratic function.
Around the solstice, the rate of change in daylight hours slows down and eventually reverses.
This change in acceleration and deceleration is why a quadratic function fits better than a linear one.
Overall, the solstice helps us understand how the fundamental rhythm of daylight hours changes as we progress through the year.
This allows for more accurate predictions and representations in modeling exercises.
It acts as the pivot point in our model, showing where the shift from increasing to decreasing daylight occurs.